\[ a (y(x)-1) y(x) y''(x)-\left ((a-1) (2 y(x)-1) y'(x)^2\right )+f(x) (y(x)-1) y(x) y'(x)=0 \] ✓ Mathematica : cpu = 0.123336 (sec), leaf count = 83
DSolve[f[x]*(-1 + y[x])*y[x]*Derivative[1][y][x] - (-1 + a)*(-1 + 2*y[x])*Derivative[1][y][x]^2 + a*(-1 + y[x])*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [a \text {$\#$1}^{-1/a} (-((\text {$\#$1}-1) \text {$\#$1}))^{\frac {1}{a}} \, _2F_1\left (\frac {1}{a},\frac {a-1}{a};1+\frac {1}{a};1-\text {$\#$1}\right )\& \right ]\left [\int _1^x\exp \left (-\int _1^{K[3]}\frac {f(K[1])}{a}dK[1]\right ) c_1dK[3]+c_2\right ]\right \}\right \}\] ✓ Maple : cpu = 0.071 (sec), leaf count = 40
dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-(a-1)*(2*y(x)-1)*diff(y(x),x)^2+f*y(x)*(-1+y(x))*diff(y(x),x)=0,y(x))
\[c_{1} {\mathrm e}^{-\frac {f x}{a}}-c_{2}+\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a} \left (\textit {\_a} -1\right )\right )^{\frac {1}{a}}}{\textit {\_a} \left (\textit {\_a} -1\right )}d \textit {\_a} = 0\]