2.519   ODE No. 519

\[ y'(x)^3-f(x) \left (a y(x)^2+b y(x)+c\right )^2=0 \]

✓ Mathematica : cpu = 7.64828 (sec), leaf count = 353

DSolve[-(f[x]*(c + b*y[x] + a*y[x]^2)^2) + Derivative[1][y][x]^3 == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x\sqrt [3]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ]\right \}\right \}\]

✓ Maple : cpu = 0.695 (sec), leaf count = 197

dsolve(diff(y(x),x)^3-f(x)*(a*y(x)^2+b*y(x)+c)^2=0,y(x))
 

\[\int _{}^{y \left (x \right )}\frac {1}{\left (a \,\textit {\_a}^{2}+\textit {\_a} b +c \right )^{{2}/{3}}}d \textit {\_a} +\int _{}^{x}-\frac {{\left (f \left (\textit {\_a} \right ) \left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{2}\right )}^{{1}/{3}}}{\left (a y \left (x \right )^{2}+b y \left (x \right )+c \right )^{{2}/{3}}}d \textit {\_a} +c_{1} = 0\]