\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) \left ( y \left ( x \right ) \right ) ^{2}}{g \left ( x \right ) }}+{\frac {{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{f \left ( x \right ) }}=0} \]
Mathematica: cpu = 26.354347 (sec), leaf count = 157 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {1}{(f(x) K[2]+g(x))^2}-\int _1^x \left (\frac {2 \left (K[2]^2 f(K[1]) f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (K[2] f(K[1])+g(K[1]))^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (K[2] f(K[1])+g(K[1]))^2}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x -\frac {y(x)^2 f(K[1]) f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (y(x) f(K[1])+g(K[1]))^2} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.343 (sec), leaf count = 57 \[ \left \{ y \left ( x \right ) =-{\frac {1}{ \left ( f \left ( x \right ) \right ) ^{2}} \left ( g \left ( x \right ) f \left ( x \right ) \int \!{ \frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x+g \left ( x \right ) f \left ( x \right ) {\it \_C1}+1 \right ) \left ( { \it \_C1}+\int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}} \,{\rm d}x \right ) ^{-1}} \right \} \]
Sage: cpu = 0.1 (sec), leaf count = 0 \[ \left [\left [\left [y\left (x\right ) = 0, -\frac {u D[0]\left (f\right )\left (x\right )^{2} D[0]\left (g\right )\left (x\right )}{f\left (x\right ) g\left (x\right )^{2}} = 0\right ]\right ], \text {\texttt {riccati}}\right ] \]