\[ y'(x)=\frac {\alpha ^3 y(x)^3+\alpha ^3 y(x)^2+\alpha ^3+3 \alpha ^2 \beta x y(x)^2+2 \alpha ^2 \beta x y(x)+3 \alpha \beta ^2 x^2 y(x)+\alpha \beta ^2 x^2+\beta ^3 x^3}{\alpha ^3} \] ✓ Mathematica : cpu = 2.47953 (sec), leaf count = 902
DSolve[Derivative[1][y][x] == (alpha^3 + alpha*beta^2*x^2 + beta^3*x^3 + 2*alpha^2*beta*x*y[x] + 3*alpha*beta^2*x^2*y[x] + alpha^3*y[x]^2 + 3*alpha^2*beta*x*y[x]^2 + alpha^3*y[x]^3)/alpha^3,y[x],x]
\[\text {Solve}\left [\frac {1}{9} \text {RootSum}\left [841 \alpha ^2 \text {$\#$1}^9+729 \beta ^2 \text {$\#$1}^9+1566 \alpha \beta \text {$\#$1}^9+2523 \alpha ^2 \text {$\#$1}^6+2187 \beta ^2 \text {$\#$1}^6+4698 \alpha \beta \text {$\#$1}^6+2496 \alpha ^2 \text {$\#$1}^3+2187 \beta ^2 \text {$\#$1}^3+4698 \alpha \beta \text {$\#$1}^3+841 \alpha ^2+729 \beta ^2+1566 \alpha \beta \& ,\frac {841 \alpha ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+729 \beta ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+1566 \alpha \beta \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^6+87 \alpha ^{5/3} \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^4+81 \alpha ^{2/3} \beta \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^4+1682 \alpha ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+1458 \beta ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+3132 \alpha \beta \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^3+9 \alpha ^{4/3} (29 \alpha +27 \beta )^{2/3} \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}^2+87 \alpha ^{5/3} \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}+81 \alpha ^{2/3} \beta \sqrt [3]{29 \alpha +27 \beta } \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right ) \text {$\#$1}+841 \alpha ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )+729 \beta ^2 \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )+1566 \alpha \beta \log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )}{841 \alpha ^2 \text {$\#$1}^8+729 \beta ^2 \text {$\#$1}^8+1566 \alpha \beta \text {$\#$1}^8+1682 \alpha ^2 \text {$\#$1}^5+1458 \beta ^2 \text {$\#$1}^5+3132 \alpha \beta \text {$\#$1}^5+832 \alpha ^2 \text {$\#$1}^2+729 \beta ^2 \text {$\#$1}^2+1566 \alpha \beta \text {$\#$1}^2}\& \right ]=\frac {1}{9} \left (\frac {29 \alpha +27 \beta }{\alpha }\right )^{2/3} x+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.063 (sec), leaf count = 42
dsolve(diff(y(x),x) = (alpha^3+y(x)^2*alpha^3+2*y(x)*alpha^2*beta*x+alpha*beta^2*x^2+y(x)^3*alpha^3+3*y(x)^2*alpha^2*beta*x+3*y(x)*alpha*beta^2*x^2+beta^3*x^3)/alpha^3,y(x))
\[y \left (x \right ) = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3} \alpha +\textit {\_a}^{2} \alpha +\alpha +\beta }d \textit {\_a} \right ) \alpha -x +c_{1}\right ) \alpha -\beta x}{\alpha }\]