\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x}{\sqrt {a}y \left ( x \right ) }F \left ( {\frac {a \left ( y \left ( x \right ) \right ) ^{2}+b{x}^{2}}{a}} \right ) }=0} \]
Mathematica: cpu = 18.189310 (sec), leaf count = 250 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \left (\frac {2 b K[1] K[2] F'\left (\frac {a K[2]^2+b K[1]^2}{a}\right )}{\sqrt {a} \left (\sqrt {a} F\left (\frac {a K[2]^2+b K[1]^2}{a}\right )+b\right )}-\frac {2 b K[1] K[2] F\left (\frac {a K[2]^2+b K[1]^2}{a}\right ) F'\left (\frac {a K[2]^2+b K[1]^2}{a}\right )}{\left (\sqrt {a} F\left (\frac {a K[2]^2+b K[1]^2}{a}\right )+b\right )^2}\right ) \, dK[1]-\frac {b K[2]}{\sqrt {a} F\left (\frac {a K[2]^2+b x^2}{a}\right )+b}\right ) \, dK[2]+\int _1^x \frac {b K[1] F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )}{\sqrt {a} \left (\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )+b\right )} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.140 (sec), leaf count = 108 \[ \left \{ y \left ( x \right ) ={\frac {1}{a}\sqrt {a \left ( -b{x}^{2}+{ \it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( \sqrt {a}b+F \left ( { \it \_a} \right ) a \right ) ^{-1}{d{\it \_a}}{a}^{{\frac {3}{2}}}b-b{x} ^{2}+2\,{\it \_C1}\,a \right ) a \right ) }},y \left ( x \right ) =-{ \frac {1}{a}\sqrt {a \left ( -b{x}^{2}+{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( \sqrt {a}b+F \left ( {\it \_a} \right ) a \right ) ^{-1}{d {\it \_a}}{a}^{{\frac {3}{2}}}b-b{x}^{2}+2\,{\it \_C1}\,a \right ) a \right ) }} \right \} \]