\[ y'(x)=\frac {F\left (\frac {x y(x)^2+1}{x}\right )}{x^2 y(x)} \] ✓ Mathematica : cpu = 0.456962 (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 = 0.165 (sec), leaf count = 72
\[\left \{y \left (x \right ) = \frac {\sqrt {\left (x \RootOf \left (c_{1} x +x \left (\int _{}^{\textit {\_Z}}\frac {1}{2 F \left (\textit {\_a} \right )-1}d \textit {\_a} \right )+1\right )-1\right ) x}}{x}, y \left (x \right ) = -\frac {\sqrt {\left (x \RootOf \left (c_{1} x +x \left (\int _{}^{\textit {\_Z}}\frac {1}{2 F \left (\textit {\_a} \right )-1}d \textit {\_a} \right )+1\right )-1\right ) x}}{x}\right \}\]