\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =1/2\, \left ( {x}^{3/2}+2\,F \left ( y \left ( x \right ) -1/6\,{x}^{3} \right ) \right ) \sqrt {x}=0} \]
Mathematica: cpu = 44.026091 (sec), leaf count = 120 \[ \text {Solve}\left [\int _1^{y(x)} -\frac {F\left (K[2]-\frac {x^3}{6}\right ) \int _1^x -\frac {K[1]^2 F'\left (K[2]-\frac {K[1]^3}{6}\right )}{2 F\left (K[2]-\frac {K[1]^3}{6}\right )^2} \, dK[1]+1}{F\left (K[2]-\frac {x^3}{6}\right )} \, dK[2]+\int _1^x \left (\frac {K[1]^2}{2 F\left (y(x)-\frac {K[1]^3}{6}\right )}+\sqrt {K[1]}\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.109 (sec), leaf count = 29 \[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\! \left ( F \left ( { \it \_a}-{\frac {{x}^{3}}{6}} \right ) \right ) ^{-1}\,{\rm d}{\it \_a} -{\frac {2}{3}{x}^{{\frac {3}{2}}}}-{\it \_C1}=0 \right \} \]