\[ \boxed { 4\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) -3\, \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+ \left ( 6\, \left ( y \left ( x \right ) \right ) ^{2}-2\,{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) y \left ( x \right ) }{f \left ( x \right ) }} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{4}-2\, \left ( y \left ( x \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +g \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{2}+f \left ( x \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.742594 (sec), leaf count = 73 \[ \text {DSolve}\left [y'(x) \left (6 y(x)^2-\frac {2 y(x) f'(x)}{f(x)}\right )+f(x) y(x)+g(x) y(x)^2+4 y(x) y''(x)-2 y(x)^2 y'(x)-3 y'(x)^2+y(x)^4=0,y(x),x\right ] \]
Maple: cpu = 0.483 (sec), leaf count = 0 \[ \text {could not solve} \]