\[ y'(x)=\frac {x (x-y(x))^3 (y(x)+x)^3}{y(x) \left (x^2-y(x)^2-1\right )} \] ✓ Mathematica : cpu = 0.177001 (sec), leaf count = 74
DSolve[Derivative[1][y][x] == (x*(x - y[x])^3*(x + y[x])^3)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x]
\[\text {Solve}\left [\frac {1}{2} \left (\text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}+1\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+x^2-y(x)^2\right )-\log \left (-\text {$\#$1}+x^2-y(x)^2\right )}{3 \text {$\#$1}^2-1}\& \right ]+x^2\right )=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.503 (sec), leaf count = 190
dsolve(diff(y(x),x) = (x-y(x))^3*(y(x)+x)^3*x/(-y(x)^2+x^2-1)/y(x),y(x))
\[\int _{\textit {\_b}}^{x}-\frac {\left (-y \left (x \right )+\textit {\_a} \right )^{3} \left (y \left (x \right )+\textit {\_a} \right )^{3} \textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} y \left (x \right )^{2}+3 y \left (x \right )^{4} \textit {\_a}^{2}-y \left (x \right )^{6}-\textit {\_a}^{2}+y \left (x \right )^{2}+1}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {\left (-\textit {\_f}^{2}+x^{2}-1\right ) \textit {\_f}}{-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}-3 \textit {\_f}^{2} x^{4}+x^{6}+\textit {\_f}^{2}-x^{2}+1}-\left (\int _{\textit {\_b}}^{x}-\frac {4 \textit {\_f} \left (-\textit {\_f} +\textit {\_a} \right )^{2} \textit {\_a} \left (\textit {\_a}^{2}-\textit {\_f}^{2}-\frac {3}{2}\right ) \left (\textit {\_f} +\textit {\_a} \right )^{2}}{\left (\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+\left (3 \textit {\_f}^{4}-1\right ) \textit {\_a}^{2}-\textit {\_f}^{6}+\textit {\_f}^{2}+1\right )^{2}}d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0\]