\[ y'(x)=\frac {F\left (\frac {x y(x)^2+1}{x}\right )}{x^2 y(x)} \] ✓ Mathematica : cpu = 25.7453 (sec), leaf count = 127
\[\text {Solve}\left [c_1=\int _1^{y(x)} \frac {\left (1-2 F\left (K[2]^2+\frac {1}{x}\right )\right ) \int _1^x \frac {2 K[2] F'\left (K[2]^2+\frac {1}{K[1]}\right )}{K[1]^2 \left (1-2 F\left (K[2]^2+\frac {1}{K[1]}\right )\right )^2} \, dK[1]+K[2]}{2 F\left (K[2]^2+\frac {1}{x}\right )-1} \, dK[2]+\int _1^x -\frac {F\left (\frac {1}{K[1]}+y(x)^2\right )}{K[1]^2 \left (2 F\left (\frac {1}{K[1]}+y(x)^2\right )-1\right )} \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 0.165 (sec), leaf count = 72
\[ \left \{ y \left ( x \right ) ={\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+x{\it \_C1}+1 \right ) x-1 \right ) }},y \left ( x \right ) =-{\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+x{\it \_C1}+1 \right ) x-1 \right ) }} \right \} \]