____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a \sqrt {x} y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (a*Sqrt[x]*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {2}{3} a x^{3/2}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde := diff(w(x,y),x)+ (a*sqrt(x)*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-2/3\,{x}^{3/2}a}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a \sqrt {x} y+ b \sqrt {y}) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (a*Sqrt[x]*y + b*Sqrt[y])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {1}{3} a x^{3/2}} \left (3^{2/3} b e^{\frac {1}{3} a x^{3/2}} \text {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )-3 a^{2/3} \sqrt {y}\right )}{3 a^{2/3}}\right )\right \},\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {1}{3} a x^{3/2}} \left (3^{2/3} b e^{\frac {1}{3} a x^{3/2}} \text {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )+3 a^{2/3} \sqrt {y}\right )}{3 a^{2/3}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde := diff(w(x,y),x)+ (a*sqrt(x)*y+b*sqrt(y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/10\,{\frac { \left ( 3\,\sqrt [3]{3} \WhittakerM \left ( 1/3,5/6,1/3\,{x}^{3/2}a \right ) {{\rm e}^{1/6\,{x}^{3/2}a}}bx+5\,bx\sqrt [3]{{x}^{3/2}a}-10\,\sqrt {y}\sqrt [3]{{x}^{3/2}a} \right ) {{\rm e}^{-1/3\,{x}^{3/2}a}}}{\sqrt [3]{{x}^{3/2}a}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +(a \sqrt {x} y+ b x \sqrt {y}) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a]; pde = D[w[x, y], x] + (a*Sqrt[x]*y + b*x*Sqrt[y])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {1}{3} a x^{3/2}} \left (\sqrt [3]{3} b e^{\frac {1}{3} a x^{3/2}} \text {Gamma}\left (\frac {1}{3},\frac {1}{3} a x^{3/2}\right )-3 a^{4/3} \sqrt {y}+3 \sqrt [3]{a} b \sqrt {x}\right )}{3 a^{4/3}}\right )\right \},\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {1}{3} a x^{3/2}} \left (\sqrt [3]{3} b e^{\frac {1}{3} a x^{3/2}} \text {Gamma}\left (\frac {1}{3},\frac {1}{3} a x^{3/2}\right )+3 a^{4/3} \sqrt {y}+3 \sqrt [3]{a} b \sqrt {x}\right )}{3 a^{4/3}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a'; pde := diff(w(x,y),x)+ (a*sqrt(x)*y+b*x*sqrt(y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/4\,{\frac {{{\rm e}^{-1/3\,{x}^{3/2}a}} \left ( 3\,\sqrt [6]{3} \WhittakerM \left ( 1/6,2/3,1/3\,{x}^{3/2}a \right ) \sqrt {x}{{\rm e}^{1/6\,{x}^{3/2}a}}b-4\,\sqrt {y}a\sqrt [6]{{x}^{3/2}a} \right ) }{\sqrt [6]{{x}^{3/2}a}a}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +A \sqrt {a x + b y+ c} w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c, A]; pde = D[w[x, y], x] + A*Sqrt[a*x + b*y + c]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 a^2 \log \left (-a e^{-\frac {\sqrt {a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}} \left (-\frac {\sqrt {a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}-1\right )\right )+a A^2 b^2 x+A^2 b^2 c}{a A^2 b^2}\right )\right \},\left \{w(x,y)\to c_1\left (\frac {2 a^2 \log \left (-a e^{\frac {\sqrt {a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}} \left (\frac {\sqrt {a A^2 b^2 x+A^2 b^3 y+A^2 b^2 c}}{a}-1\right )\right )+a A^2 b^2 x+A^2 b^2 c}{a A^2 b^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';A:='A'; C:='C';b:='b'; pde := diff(w(x,y),x)+ A*sqrt(a*x+b*y+c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {x{A}^{2}{b}^{2}-2\,A\sqrt {ax+by+c}b+a\ln \left ( {A}^{2}a{b}^{2}x+{A}^{2}{b}^{3}y+{A}^{2}{b}^{2}c-{a}^{2} \right ) -a\ln \left ( A\sqrt {ax+by+c}b-a \right ) +a\ln \left ( A\sqrt {ax+by+c}b+a \right ) }{{A}^{2}{b}^{2}}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a y + b \sqrt {y^2+c x^2} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, a, b]; pde = x*D[w[x, y], x] + (a*y + b*Sqrt[y^2 + c*x^2])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b'; pde := x*diff(w(x,y),x)+ ( a*y + b *sqrt(y^2+c*x^2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \] Timed out
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \left (a x + b \sqrt {y} \right ) w_x - \left ( c \sqrt {x} + a y \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, a, b, c]; pde = (a*x + b*Sqrt[y])*D[w[x, y], x] - (c*Sqrt[x] + a*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {9 a^3 x^3+16 b^2 c x^{3/2}}{24 b^2}\right )\right \},\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (3 a x y-2 b y^{3/2}+2 c x^{3/2}\right )\right )\right \},\left \{w(x,y)\to c_1\left (\frac {1}{3} \left (3 a x y+2 b y^{3/2}+2 c x^{3/2}\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';c:='c'; pde := (a*x+b*sqrt(y))* diff(w(x,y),x)- (c*sqrt(x)+a*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \RootOf \left ( 8\,{y}^{5/2}ab{c}^{2}+3\,{y}^{4}{a}^{4}-2\,\sqrt [3]{-4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c}{a}^{3}{y}^{3}+3\, \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}{a}^{2}{y}^{2}-4\,x{c}^{2} \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{2/3}+12\,a{c}^{2}y{\it \_Z}-4\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}acy+ \left ( -4\,{y}^{3/2}b{c}^{2}-{y}^{3}{a}^{3}-6\,{c}^{2}{\it \_Z}+2\,\sqrt {4\,{y}^{3}{b}^{2}{c}^{2}+2\,{y}^{9/2}{a}^{3}b+12\,{y}^{3/2}{\it \_Z}\,b{c}^{2}+3\,{\it \_Z}\,{a}^{3}{y}^{3}+9\,{{\it \_Z}}^{2}{c}^{2}}c \right ) ^{4/3} \right ) \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \sqrt {f(x)} w_x + \sqrt {f(y)} w_y = 0 \] Where \(f(t) = \sum _{n=0}^{4} a_n t^n \)
Mathematica ✗
ClearAll[w, x, y, t, n, a]; f[t_] := Sum[a[n]*t^n, {n, 1, 4}]; pde = Sqrt[f[x]]*D[w[x, y], x] + Sqrt[f[y]]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';n:='n';t:='t'; f:=t->sum(a[n]*t^n,n=1..4); pde := sqrt(f(x))* diff(w(x,y),x)+ sqrt(f(y))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{1 \left ( 1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \sqrt {{x \left ( -1/4\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{-1}}} \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{2}\sqrt {{1 \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{-1}}}\sqrt {{1 \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{-1}}}\EllipticF \left ( \sqrt {{x \left ( -1/4\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( -1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{-1}}},\sqrt {{1 \left ( 1/4\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( 1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( 1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( 1/4\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1}}} \right ) \left ( -1/4\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) ^{-1} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}-1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) ^{-1}{\frac {1}{\sqrt {a_{{4}}x \left ( x-1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}} \right ) \left ( x+1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}-i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) \left ( x+1/12\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}-1/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}}+1/3\,{\frac {a_{{3}}}{a_{{4}}}}+i/2\sqrt {3} \left ( 1/6\,{\frac {\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{a_{{4}}}}+2/3\,{\frac {3\,a_{{2}}a_{{4}}-{a_{{3}}}^{2}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}} \right ) \right ) }}}}-8\,{\frac { \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) ^{2}\sqrt {6}}{a_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}} \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) \left ( \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }\sqrt {-6\,{\frac { \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{ \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }}}\sqrt {{\frac { \left ( \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}-12\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}- \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }}}\sqrt {{\frac { \left ( \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+12\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }}}\EllipticF \left ( \sqrt {-6\,{\frac { \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}}{ \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }}},\sqrt {{\frac { \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}-3\, \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+36\,a_{{2}}a_{{4}}-12\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{ \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}- \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+3\, \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-36\,a_{{2}}a_{{4}}+12\,{a_{{3}}}^{2} \right ) }}} \right ) {\frac {1}{\sqrt {{\frac {y \left ( -6\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-2\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}-12\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}- \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}-4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+12\,a_{{2}}a_{{4}}-4\,{a_{{3}}}^{2} \right ) \left ( i \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}\sqrt {3}+12\,i\sqrt {3}a_{{2}}a_{{4}}-4\,i\sqrt {3}{a_{{3}}}^{2}+12\,ya_{{4}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}+ \left ( 12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3} \right ) ^{2/3}+4\,a_{{3}}\sqrt [3]{12\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-108\,a_{{1}}{a_{{4}}}^{2}+36\,a_{{2}}a_{{3}}a_{{4}}-8\,{a_{{3}}}^{3}}-12\,a_{{2}}a_{{4}}+4\,{a_{{3}}}^{2} \right ) }{{a_{{4}}}^{2} \left ( 3\,\sqrt {3}\sqrt {27\,{a_{{1}}}^{2}{a_{{4}}}^{2}-18\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+4\,a_{{1}}{a_{{3}}}^{3}+4\,{a_{{2}}}^{3}a_{{4}}-{a_{{2}}}^{2}{a_{{3}}}^{2}}a_{{4}}-27\,a_{{1}}{a_{{4}}}^{2}+9\,a_{{2}}a_{{3}}a_{{4}}-2\,{a_{{3}}}^{3} \right ) }}}}}} \right ) \]