\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-{\frac {y \left ( x \right ) \left ( xy \left ( x \right ) +1 \right ) }{x \left ( xy \left ( x \right ) +1-y \left ( x \right ) \right ) }}=0} \]
Mathematica: cpu = 13.841758 (sec), leaf count = 397 \[ \text {Solve}\left [\frac {\sqrt [3]{-2} \left (\frac {2^{2/3} ((x-1) y(x)-2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right ) \left (\frac {-x y(x)+y(x)+2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right ) \left (\left (\frac {\sqrt [3]{-1} (-x y(x)+y(x)+2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+1\right ) \log \left (\frac {2^{2/3} ((x-1) y(x)-2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right )-\left (\frac {\sqrt [3]{-1} (-x y(x)+y(x)+2)}{\sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+1\right ) \log \left (\frac {-x y(x)+y(x)+2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(x-1)^3}} (x-1) ((x-1) y(x)+1)}+(-2)^{2/3}\right )-3\right )}{9 \left (\frac {((x-1) y(x)-2)^3}{((x-1) y(x)+1)^3}+\frac {3 \sqrt [3]{-1} ((x-1) y(x)-2)}{\left (-\frac {1}{(x-1)^3}\right )^{4/3} (x-1)^4 ((x-1) y(x)+1)}+2\right )}=c_1+\frac {1}{9} 2^{2/3} \left (-\frac {1}{(x-1)^3}\right )^{2/3} (x-1)^2 (\log (1-x)-\log (x)),y(x)\right ] \]
Maple: cpu = 0.078 (sec), leaf count = 32 \[ \left \{ y \left ( x \right ) =-2\,{\frac {1}{x}{{\rm e}^{-{\it lambertW } \left ( -2\,{\frac { \left ( x-1 \right ) \left ( {{\rm e}^{{\it \_C1}} } \right ) ^{3}{{\rm e}^{-1}}}{x}} \right ) +3\,{\it \_C1}-1}}} \right \} \]