\[ y'(x)-\left (\frac {\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3}{\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3}\right )^{2/3}=0 \] ✓ Mathematica : cpu = 1.49231 (sec), leaf count = 733
\[\text {Solve}\left [\frac {3 (\text {a0}+y(x) (\text {a1}+y(x) (\text {a2}+\text {a3} y(x))))^{2/3} \left (y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]\right ) F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]}\right )}{5 \left (\frac {y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \left (\frac {y(x)-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}\right )^{2/3}}=\frac {3 (\text {a0}+x (\text {a1}+x (\text {a2}+\text {a3} x)))^{2/3} \left (x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]\right ) F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-x}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]},\frac {\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-x}{\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]}\right )}{5 \left (\frac {x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \left (\frac {x-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\& ,3\right ]}\right )^{2/3}}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.469 (sec), leaf count = 91
\[\int _{}^{y \relax (x )}\left (\textit {\_a}^{3} \mathit {a3} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a} \mathit {a1} +\mathit {a0} \right )^{\frac {2}{3}}d \textit {\_a} +\int _{}^{x}-\left (\frac {\textit {\_a}^{3} \mathit {a3} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a} \mathit {a1} +\mathit {a0}}{\mathit {a3} y \relax (x )^{3}+\mathit {a2} y \relax (x )^{2}+\mathit {a1} y \relax (x )+\mathit {a0}}\right )^{\frac {2}{3}} \left (\mathit {a3} y \relax (x )^{3}+\mathit {a2} y \relax (x )^{2}+\mathit {a1} y \relax (x )+\mathit {a0} \right )^{\frac {2}{3}}d \textit {\_a} +c_{1} = 0\]