\[ y'(x)=\frac {F\left (x^2+y(x)^2\right )-x}{y(x)} \] ✓ Mathematica : cpu = 28.1177 (sec), leaf count = 92
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (-\int _1^x \frac {2 K[1] K[2] F'\left (K[1]^2+K[2]^2\right )}{F\left (K[1]^2+K[2]^2\right )^2} \, dK[1]-\frac {K[2]}{F\left (K[2]^2+x^2\right )}\right ) \, dK[2]+\int _1^x \left (1-\frac {K[1]}{F\left (K[1]^2+y(x)^2\right )}\right ) \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 0.115 (sec), leaf count = 57
\[ \left \{ y \left ( x \right ) =\sqrt {-{x}^{2}+{\it RootOf} \left ( -2\,x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) },y \left ( x \right ) =-\sqrt {-{x}^{2}+{\it RootOf} \left ( -2\,x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) } \right \} \]