\[ \boxed { {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) +3\,f \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) +2\, \left ( f \left ( x \right ) \right ) ^{2}+4\,g \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( 4\,f \left ( x \right ) g \left ( x \right ) +2\,{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.015002 (sec), leaf count = 55 \[ \text {DSolve}\left [y'(x) \left (f'(x)+2 f(x)^2+4 g(x)\right )+y(x) \left (4 f(x) g(x)+2 g'(x)\right )+3 f(x) y''(x)+y^{(3)}(x)=0,y(x),x\right ] \]
Maple: cpu = 0.110 (sec), leaf count = 30 \[ \left \{ y \left ( x \right ) = \left ( {\it DESol} \left ( \left \{ { \frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +f \left ( x \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) +g \left ( x \right ) {\it \_Y} \left ( x \right ) \right \} , \left \{ { \it \_Y} \left ( x \right ) \right \} \right ) \right ) ^{2} \right \} \]