\[ y'(x)=\frac {x \left (x^2 y(x)^3+\left (x^2+1\right )^{3/2} y(x)^2+x^2 \left (x^2+1\right )^{3/2}+\left (x^2+1\right )^{3/2}+y(x)^3\right )}{\left (x^2+1\right )^3} \] ✓ Mathematica : cpu = 0.496181 (sec), leaf count = 148
DSolve[Derivative[1][y][x] == (x*((1 + x^2)^(3/2) + x^2*(1 + x^2)^(3/2) + (1 + x^2)^(3/2)*y[x]^2 + y[x]^3 + x^2*y[x]^3))/(1 + x^2)^3,y[x],x]
\[\text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {$\#$1}^3+6 \sqrt [3]{38} \text {$\#$1}-19\& ,\frac {\log \left (\frac {\frac {3 x y(x)}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}-\text {$\#$1}\right )}{2 \sqrt [3]{38}-19 \text {$\#$1}^2}\& \right ]=\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.1 (sec), leaf count = 48
dsolve(diff(y(x),x) = ((x^2+1)^(3/2)*x^2+(x^2+1)^(3/2)+y(x)^2*(x^2+1)^(3/2)+x^2*y(x)^3+y(x)^3)*x/(x^2+1)^3,y(x))
\[y \left (x \right ) = \frac {\sqrt {x^{2}+1}\, \left (19 \RootOf \left (-1296 \left (\int _{}^{\textit {\_Z}}\frac {1}{361 \textit {\_a}^{3}-432 \textit {\_a} +432}d \textit {\_a} \right )+2 \ln \left (x^{2}+1\right )+3 c_{1}\right )-6\right )}{18}\]