\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =2\,{ \left ( y \left ( x \right ) \right ) ^{3} \left ( 1+2\,F \left ( {\frac {1+4\,x \left ( y \left ( x \right ) \right ) ^{2}}{ \left ( y \left ( x \right ) \right ) ^{2}}} \right ) y \left ( x \right ) \right ) ^{-1}}=0} \]
Mathematica: cpu = 26.277337 (sec), leaf count = 140 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \frac {\left (\frac {8 K[1]}{K[2]}-\frac {2 \left (4 K[1] K[2]^2+1\right )}{K[2]^3}\right ) F'\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )}{F\left (\frac {4 K[1] K[2]^2+1}{K[2]^2}\right )^2} \, dK[1]+\frac {1}{2 K[2]^3 F\left (\frac {4 x K[2]^2+1}{K[2]^2}\right )}+\frac {1}{K[2]^2}\right ) \, dK[2]+\int _1^x -\frac {1}{F\left (\frac {4 y(x)^2 K[1]+1}{y(x)^2}\right )} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.141 (sec), leaf count = 30 \[ \left \{ -{\it \_C1}- \left ( y \left ( x \right ) \right ) ^{-1}-{\frac {\int ^{4\,x+ \left ( y \left ( x \right ) \right ) ^{-2}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}}{4}}=0 \right \} \]