\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-{\frac {-{x}^{2}+2\,{x}^{3}y \left ( x \right ) -F \left ( \left ( xy \left ( x \right ) -1 \right ) x \right ) }{{x}^{4}}}=0} \]
Mathematica: cpu = 52.212630 (sec), leaf count = 174 \[ \text {Solve}\left [\int _1^{y(x)} -\frac {F(x (x K[2]-1)) \int _1^x \left (\frac {2 K[2] K[1]^3 F'(K[1] (K[1] K[2]-1))}{F(K[1] (K[1] K[2]-1))^2}-\frac {K[1]^2 F'(K[1] (K[1] K[2]-1))}{F(K[1] (K[1] K[2]-1))^2}-\frac {2 K[1]}{F(K[1] (K[1] K[2]-1))}\right ) \, dK[1]+x^2}{F(x (x K[2]-1))} \, dK[2]+\int _1^x \left (-\frac {2 y(x) K[1]}{F(K[1] (y(x) K[1]-1))}+\frac {1}{F(K[1] (y(x) K[1]-1))}+\frac {1}{K[1]^2}\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.093 (sec), leaf count = 26 \[ \left \{ y \left ( x \right ) ={\frac {{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+ {\it \_C1}\,x+1 \right ) +x}{{x}^{2}}} \right \} \]