\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {-x+F \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) }{y \left ( x \right ) }}=0} \]
Mathematica: cpu = 22.629374 (sec), leaf count = 92 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \frac {2 K[1] K[2] F'\left (K[1]^2+K[2]^2\right )}{F\left (K[1]^2+K[2]^2\right )^2} \, dK[1]-\frac {K[2]}{F\left (K[2]^2+x^2\right )}\right ) \, dK[2]+\int _1^x \left (1-\frac {K[1]}{F\left (K[1]^2+y(x)^2\right )}\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.093 (sec), leaf count = 57 \[ \left \{ y \left ( x \right ) =\sqrt {-{x}^{2}+{\it RootOf} \left ( -2\,x +\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{ d{\it \_a}}+2\,{\it \_C1} \right ) },y \left ( x \right ) =-\sqrt {-{x}^{ 2}+{\it RootOf} \left ( -2\,x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) } \right \} \]