#### 2.825   ODE No. 825

$y'(x)=\frac {x \left (x^2 y(x)^3+\left (x^2+1\right )^{3/2} y(x)^2+x^2 \left (x^2+1\right )^{3/2}+\left (x^2+1\right )^{3/2}+y(x)^3\right )}{\left (x^2+1\right )^3}$ Mathematica : cpu = 0.412246 (sec), leaf count = 148

$\text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {\#1}^3+6 \sqrt [3]{38} \text {\#1}-19\& ,\frac {\log \left (\frac {\frac {3 x y(x)}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}-\text {\#1}\right )}{2 \sqrt [3]{38}-19 \text {\#1}^2}\& \right ]=c_1+\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2},y(x)\right ]$ Maple : cpu = 0.075 (sec), leaf count = 48

$\left \{ y \left ( x \right ) ={\frac {19\,{\it RootOf} \left ( -1296\,\int ^{{\it \_Z}}\! \left ( 361\,{{\it \_a}}^{3}-432\,{\it \_a}+432 \right ) ^{-1}{d{\it \_a}}+2\,\ln \left ( {x}^{2}+1 \right ) +3\,{\it \_C1} \right ) -6}{18}\sqrt {{x}^{2}+1}} \right \}$