\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {{\alpha }^{3}+ \left ( y \left ( x \right ) \right ) ^{2}{\alpha }^{3}+2\,y \left ( x \right ) {\alpha }^{2}\beta \,x+\alpha \,{\beta }^{2}{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{3}{\alpha }^{3}+3\, \left ( y \left ( x \right ) \right ) ^{2}{\alpha }^{2}\beta \,x+3\,y \left ( x \right ) \alpha \,{\beta }^{2}{x}^{2}+{\beta }^{3}{x}^{3}}{{\alpha }^{3}}}=0} \]
Mathematica: cpu = 0.149019 (sec), leaf count = 145 \[ \text {Solve}\left [-\frac {1}{3} (29 \alpha +27 \beta )^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (29 \alpha +27 \beta )^{2/3}-3 \text {$\#$1} \alpha ^{2/3}+(29 \alpha +27 \beta )^{2/3}\& ,\frac {\log \left (\frac {\frac {\alpha +3 \beta x}{\alpha }+3 y(x)}{\sqrt [3]{\frac {29 \alpha +27 \beta }{\alpha }}}-\text {$\#$1}\right )}{\alpha ^{2/3}-\text {$\#$1}^2 (29 \alpha +27 \beta )^{2/3}}\& \right ]=\frac {1}{9} x \left (\frac {29 \alpha +27 \beta }{\alpha }\right )^{2/3}+c_1,y(x)\right ] \]
Maple: cpu = 0.047 (sec), leaf count = 42 \[ \left \{ y \left ( x \right ) ={\frac {{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( {{\it \_a}}^{3}\alpha +{{\it \_a}}^{2}\alpha +\alpha + \beta \right ) ^{-1}{d{\it \_a}}\alpha -x+{\it \_C1} \right ) \alpha - \beta \,x}{\alpha }} \right \} \]