\[ y'(x)=\frac {F\left (\frac {x y(x)^2+1}{x}\right )}{x^2 y(x)} \] ✓ Mathematica : cpu = 24.2109 (sec), leaf count = 201
\[\text {Solve}\left [\int _1^{y(x)} \left (\frac {K[2]}{2 F\left (\frac {x K[2]^2+1}{x}\right )-1}-\int _1^x \left (\frac {4 K[2] F\left (\frac {K[1] K[2]^2+1}{K[1]}\right ) F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{K[1]^2 \left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right )^2}-\frac {2 K[2] F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{K[1]^2 \left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right )}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x -\frac {F\left (\frac {y(x)^2 K[1]+1}{K[1]}\right )}{K[1]^2 \left (2 F\left (\frac {y(x)^2 K[1]+1}{K[1]}\right )-1\right )} \, dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.149 (sec), leaf count = 72
\[ \left \{ y \left ( x \right ) ={\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+x{\it \_C1}+1 \right ) x-1 \right ) }},y \left ( x \right ) =-{\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+x{\it \_C1}+1 \right ) x-1 \right ) }} \right \} \]