\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( 1+2\,y \left ( x \right ) \right ) \left ( 1+y \left ( x \right ) \right ) }{x \left ( -2\,y \left ( x \right ) -2+x+2\,xy \left ( x \right ) \right ) }}=0} \]
Mathematica: cpu = 1.137644 (sec), leaf count = 127 \[ \text {Solve}\left [\frac {2^{2/3} \left (x \log \left (-\frac {y(x)+1}{2 (x-1) y(x)+x-2}\right )-x \log \left (\frac {2 x y(x)+x}{2 (x-1) y(x)+x-2}\right )+2 x y(x) \left (\log \left (-\frac {y(x)+1}{2 (x-1) y(x)+x-2}\right )-\log \left (\frac {2 x y(x)+x}{2 (x-1) y(x)+x-2}\right )+\log (x)+1\right )+x+x \log (x)-1\right )}{9 (2 x y(x)+x)}=c_1,y(x)\right ] \]
Maple: cpu = 0.062 (sec), leaf count = 44 \[ \left \{ y \left ( x \right ) =-{\frac {1}{2} \left ( x{\it lambertW} \left ( {\frac {1}{x{{\rm e}^{{x}^{-1}}}{\it \_C1}}} \right ) +2 \right ) \left ( x{\it lambertW} \left ( {\frac {1}{x{{\rm e}^{{x}^{-1} }}{\it \_C1}}} \right ) +1 \right ) ^{-1}} \right \} \]