\[ \boxed { f \left ( {x}^{2}+a \left ( y \left ( x \right ) \right ) ^{2} \right ) \left ( ay \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +x \right ) -y \left ( x \right ) -x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =0} \]
Mathematica: cpu = 264.973147 (sec), leaf count = 88 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \left (1-2 a K[1] K[2] f'\left (a K[2]^2+K[1]^2\right )\right ) \, dK[1]-a K[2] f\left (a K[2]^2+x^2\right )+x\right ) \, dK[2]+\int _1^x \left (y(x)-K[1] f\left (K[1]^2+a y(x)^2\right )\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.046 (sec), leaf count = 45 \[ \left \{ -{ax \left ( y \left ( x \right ) \right ) ^{2}{\frac {1}{\sqrt {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}}}}}-\int ^{-{\frac {a \left ( y \left ( x \right ) \right ) ^{2}}{2}}-{\frac {{x}^{2}}{2}}}\!f \left ( -2\,{\it \_a} \right ) {d{\it \_a}}+{\it \_C1}=0 \right \} \]