\[ y'(x)=\frac {F(x (x y(x)-1))-2 x^3 y(x)+x^2}{x^4} \] ✓ Mathematica : cpu = 55.7055 (sec), leaf count = 126
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (-\int _1^x \frac {K[1] \left (K[1] (2 K[1] K[2]-1) F'(K[1] (K[1] K[2]-1))-2 F(K[1] (K[1] K[2]-1))\right )}{F(K[1] (K[1] K[2]-1))^2} \, dK[1]-\frac {x^2}{F(x (x K[2]-1))}\right ) \, dK[2]+\int _1^x \left (\frac {1-2 y(x) K[1]}{F(K[1] (y(x) K[1]-1))}+\frac {1}{K[1]^2}\right ) \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 1.794 (sec), leaf count = 26
\[ \left \{ y \left ( x \right ) ={\frac {{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+{\it \_C1}\,x+1 \right ) +x}{{x}^{2}}} \right \} \]