#### 2.852   ODE No. 852

$y'(x)=\frac {\alpha ^3 y(x)^3+\alpha ^3 y(x)^2+\alpha ^3+3 \alpha ^2 \beta x y(x)^2+2 \alpha ^2 \beta x y(x)+3 \alpha \beta ^2 x^2 y(x)+\alpha \beta ^2 x^2+\beta ^3 x^3}{\alpha ^3}$ Mathematica : cpu = 0.32207 (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

$\left \{ y \left ( x \right ) ={\frac {{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( {{\it \_a}}^{3}\alpha +{{\it \_a}}^{2}\alpha +\alpha +\beta \right ) ^{-1}{d{\it \_a}}\alpha -x+{\it \_C1} \right ) \alpha -\beta \,x}{\alpha }} \right \}$