\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) - \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-f \left ( x \right ) y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -g \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 10.804372 (sec), leaf count = 70 \[ \left \{\left \{y(x)\to c_2 \exp \left (\int _1^x \left (c_1 e^{\int _1^{K[3]} f(K[1]) \, dK[1]}+e^{\int _1^{K[3]} f(K[1]) \, dK[1]} \int _1^{K[3]} g(K[2]) e^{-\int _1^{K[2]} f(K[1]) \, dK[1]} \, dK[2]\right ) \, dK[3]\right )\right \}\right \} \]
Maple: cpu = 0.063 (sec), leaf count = 61 \[ \left \{ y \left ( x \right ) ={\frac {{\it \_C2}}{{{\rm e}^{{\it \_C1} \,\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x}}} {{\rm e}^{\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}} \,{\rm d}x\int \!{\frac {g \left ( x \right ) }{{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}}}\,{\rm d}x}} \left ( {{\rm e}^{\int \!{ \frac {\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}} \,{\rm d}xg \left ( x \right ) }{{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}}}\,{\rm d}x}} \right ) ^{-1}} \right \} \]