\[ -\frac {a y(x)^3 f'(x)}{a+2}+\frac {a f(x)^2 y(x)^4}{(a+2)^2}-\frac {(a-1) y'(x)^2}{a}-f(x) y(x)^2 y'(x)+y(x) y''(x)=0 \] ✓ Mathematica : cpu = 21.1075 (sec), leaf count = 41
DSolve[(a*f[x]^2*y[x]^4)/(2 + a)^2 - (a*y[x]^3*Derivative[1][f][x])/(2 + a) - f[x]*y[x]^2*Derivative[1][y][x] - ((-1 + a)*Derivative[1][y][x]^2)/a + y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {(a+2) (x+c_1){}^a}{a \int _1^xf(K[3]) (c_1+K[3]){}^adK[3]+c_2}\right \}\right \}\] ✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve(diff(diff(y(x),x),x)*y(x)-(a-1)/a*diff(y(x),x)^2-f(x)*y(x)^2*diff(y(x),x)+a/(a+2)^2*f(x)^2*y(x)^4-a/(a+2)*diff(f(x),x)*y(x)^3=0,y(x))
, could not solve
dsolve(diff(diff(y(x),x),x)*y(x)-(a-1)/a*diff(y(x),x)^2-f(x)*y(x)^2*diff(y(x),x)+a/(a+2)^2*f(x)^2*y(x)^4-a/(a+2)*diff(f(x),x)*y(x)^3=0,y(x))