\[ y'(x)=\frac {F\left (\frac {x y(x)^2+1}{x}\right )}{x^2 y(x)} \] ✓ Mathematica : cpu = 0.325125 (sec), leaf count = 204
\[\text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{2 F\left (\frac {x K[2]^2+1}{x}\right )-1}-\int _1^x\left (\frac {4 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right ) K[2] F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right )^2 K[1]^2}-\frac {2 K[2] F'\left (\frac {K[1] K[2]^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] K[2]^2+1}{K[1]}\right )-1\right ) K[1]^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F\left (\frac {K[1] y(x)^2+1}{K[1]}\right )}{\left (2 F\left (\frac {K[1] y(x)^2+1}{K[1]}\right )-1\right ) K[1]^2}dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.302 (sec), leaf count = 72
\[ \left \{ y \left ( x \right ) ={\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+{\it \_C1}\,x+1 \right ) x-1 \right ) }},y \left ( x \right ) =-{\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+{\it \_C1}\,x+1 \right ) x-1 \right ) }} \right \} \]