\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-1/2\, \left ( a{x}^{2}-2\,F \left ( y \left ( x \right ) +1/8\,a{x}^{4} \right ) \right ) x=0} \]
Mathematica: cpu = 41.455764 (sec), leaf count = 123 \[ \text {Solve}\left [\int _1^{y(x)} -\frac {F\left (K[2]+\frac {a x^4}{8}\right ) \int _1^x \frac {a K[1]^3 F'\left (\frac {1}{8} a K[1]^4+K[2]\right )}{2 F\left (\frac {1}{8} a K[1]^4+K[2]\right )^2} \, dK[1]+1}{F\left (K[2]+\frac {a x^4}{8}\right )} \, dK[2]+\int _1^x \left (K[1]-\frac {a K[1]^3}{2 F\left (\frac {1}{8} a K[1]^4+y(x)\right )}\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.125 (sec), leaf count = 31 \[ \left \{ y \left ( x \right ) =-{\frac {a{x}^{4}}{8}}+{\it RootOf} \left ( -{x}^{2}+2\,\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) \right \} \]