\[ \boxed { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +f \left ( x \right ) {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) + \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +g \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( {\frac {\rm d}{{\rm d}x}}h \left ( x \right ) \right ) y \left ( x \right ) +h \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +A \left ( x \right ) \left ( f \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +g \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +h \left ( x \right ) y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 0.034004 (sec), leaf count = 76 \[ \text {DSolve}\left [A(x) \left (f(x) y''(x)+g(x) y'(x)+h(x) y(x)\right )+f'(x) y''(x)+f(x) y^{(3)}(x)+g'(x) y'(x)+g(x) y''(x)+y(x) h'(x)+h(x) y'(x)=0,y(x),x\right ] \]
Maple: cpu = 0.125 (sec), leaf count = 70 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ f \left ( x \right ) {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}{\it \_Y} \left ( x \right ) + \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) +g \left ( x \right ) +A \left ( x \right ) f \left ( x \right ) \right ) { \frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) + \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) +h \left ( x \right ) +A \left ( x \right ) g \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) + \left ( {\frac {\rm d}{ {\rm d}x}}h \left ( x \right ) +A \left ( x \right ) h \left ( x \right ) \right ) {\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]