\[ \boxed { ay \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( a-1 \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+ \left ( a+2 \right ) f \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( f \left ( x \right ) \right ) ^{2} \left ( y \left ( x \right ) \right ) ^{4}+a \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) \left ( y \left ( x \right ) \right ) ^{3}=0} \]
Mathematica: cpu = 0.827105 (sec), leaf count = 59 \[ \text {DSolve}\left [(a+2) f(x) y(x)^2 y'(x)+a y(x) y''(x)+a y(x)^3 y'(x)-(a-1) y'(x)^2+f(x)^2 y(x)^4=0,y(x),x\right ] \]
Maple: cpu = 1.123 (sec), leaf count = 0 \[ \text {could not solve} \]