\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {1}{{x}^{2}y \left ( x \right ) }F \left ( {\frac {x \left ( y \left ( x \right ) \right ) ^{2}+1}{x}} \right ) }=0} \]
Mathematica: cpu = 19.349457 (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 = 0.124 (sec), leaf count = 72 \[ \left \{ y \left ( x \right ) ={\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}x+{\it \_C1}\,x+1 \right ) x-1 \right ) }},y \left ( x \right ) =-{\frac {1}{x}\sqrt {x \left ( {\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( -1+2\,F \left ( {\it \_a} \right ) \right ) ^ {-1}{d{\it \_a}}x+{\it \_C1}\,x+1 \right ) x-1 \right ) }} \right \} \]