\[ y'(x)=\frac {-x^3+3 x^2 y(x)+x^2-3 x y(x)^2+y(x)^3}{(x-1) (x+1)} \] ✓ Mathematica : cpu = 1.5195 (sec), leaf count = 238
DSolve[Derivative[1][y][x] == (x^2 - x^3 + 3*x^2*y[x] - 3*x*y[x]^2 + y[x]^3)/((-1 + x)*(1 + x)),y[x],x]
\[\text {Solve}\left [\frac {\arctan \left (\frac {\frac {2 \left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )-\frac {1}{6} \log \left (\frac {\left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )^2}{9 \left (\frac {1}{(x-1)^3 (x+1)^3}\right )^{2/3}}-\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )=\frac {1}{2} \left (\frac {1}{\left (x^2-1\right )^3}\right )^{2/3} \left (x^2-1\right )^2 (\log (1-x)-\log (x+1))+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.727 (sec), leaf count = 187
dsolve(diff(y(x),x) = (y(x)^3-3*x*y(x)^2+3*x^2*y(x)-x^3+x^2)/(x-1)/(1+x),y(x))
\[y \left (x \right ) = \frac {\left (\left (x^{2}-1\right ) \left (3 \tan \left (\operatorname {RootOf}\left (9 \ln \left (\frac {1+x}{x -1}\right ) \left (\frac {1}{\left (1+x \right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} x^{4}-18 \ln \left (\frac {1+x}{x -1}\right ) \left (\frac {1}{\left (1+x \right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} x^{2}+9 \ln \left (\frac {1+x}{x -1}\right ) \left (\frac {1}{\left (1+x \right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}}+6 \sqrt {3}\, \textit {\_Z} +2 \ln \left (\frac {3 \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{3} \sqrt {3}}{8 \left (1+x \right )^{3} \left (x -1\right )^{3}}\right )+3 \ln \left (\frac {4}{3 \left (\tan \left (\textit {\_Z} \right )^{2}+1\right )}\right )-2 \ln \left (\frac {1}{\left (1+x \right )^{3} \left (x -1\right )^{3}}\right )-18 c_{1}\right )\right )+\sqrt {3}\right ) \left (\frac {1}{\left (1+x \right )^{3} \left (x -1\right )^{3}}\right )^{\frac {1}{3}}+2 \sqrt {3}\, x \right ) \sqrt {3}}{6}\]