\[ y(x)^2 \left (a y(x)^3+1\right )+2 y(x) y''(x)-6 y'(x)^2=0 \] ✓ Mathematica : cpu = 17.5864 (sec), leaf count = 2761
DSolve[y[x]^2*(1 + a*y[x]^3) - 6*Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\text {Solve}\left [-\frac {4 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right ),-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\operatorname {EllipticPi}\left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )},\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right ),-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}} \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right ),-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\operatorname {EllipticPi}\left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )},\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}}\right ),-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right )}} \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.094 (sec), leaf count = 71
dsolve(2*diff(diff(y(x),x),x)*y(x)-6*diff(y(x),x)^2+(1+a*y(x)^3)*y(x)^2=0,y(x))
\[\int _{}^{y \left (x \right )}-\frac {2}{\sqrt {4 \textit {\_a}^{4} c_{1}+4 \textit {\_a}^{3} a +1}\, \textit {\_a}}d \textit {\_a} -x -c_{2} = 0\]