\[ \left (x^2+1\right ) y'(x)+(2 x y(x)-1) \left (y(x)^2+1\right )=0 \] ✓ Mathematica : cpu = 0.567877 (sec), leaf count = 161
\[\text {Solve}\left [c_1=\frac {i \left (x \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )-2\right )+y(x) \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )+2 x^2\right )\right )}{2 (x y(x)-1) \sqrt [4]{-\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}}},y(x)\right ]\]
✓ Maple : cpu = 0.059 (sec), leaf count = 85
\[ \left \{ {\it \_C1}+{x{\frac {1}{\sqrt [4]{ \left ( {x}^{-1}+{{x}^{2} \left ( {\frac {y \left ( x \right ) {x}^{4}}{{x}^{2}+1}}-{\frac {{x}^{3}}{{x}^{2}+1}} \right ) ^{-1}} \right ) ^{2}+1}}}}+{\frac {y \left ( x \right ) +x}{2\,xy \left ( x \right ) -2}{\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{4}};\,{\frac {3}{2}};\,-{\frac { \left ( y \left ( x \right ) +x \right ) ^{2}}{ \left ( xy \left ( x \right ) -1 \right ) ^{2}}})}}=0 \right \} \]