\[ \boxed { y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +f \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) g \left ( x \right ) +x=0} \]
Mathematica: cpu = 22.411346 (sec), leaf count = 92 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {K[2]}{f\left (K[2]^2+x^2\right )}-\int _1^x -\frac {2 K[1] K[2] f'\left (K[1]^2+K[2]^2\right )}{f\left (K[1]^2+K[2]^2\right )^2} \, dK[1]\right ) \, dK[2]+\int _1^x \left (\frac {K[1]}{f\left (K[1]^2+y(x)^2\right )}+g(K[1])\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.094 (sec), leaf count = 30 \[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{\frac {{\it \_a}}{f \left ( {{\it \_a}}^{2}+{x}^{2} \right ) }}\,{\rm d}{\it \_a}+\int \!g \left ( x \right ) \,{\rm d}x-{\it \_C1}=0 \right \} \]