Mathematica 10.3 evaluates

Integrate[((a+b x)^m (c+d x)^n) / (e+f x)^(m+n+2),x]

to $$\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-2} \left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \left(\frac{b (e+f x)}{b e-a f}\right)^{n+1} \, _2F_1\left(m+1,-n;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right)}{b (m+1)}$$ which is incorrect, in the sense that its derivative is not equal to the integrand. This can be verified by noting that the substitution

Simplify[D[Integrate[((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 2), x], x]] /.
     {a -> 2, b -> -2, c -> -2, d -> 4, e -> 7, f -> -4, m -> 1, n -> 1/2, x -> 2}

evaluates to $2 i \sqrt{6}$; whereas

((a + b*x)^m*(c + d*x)^n)/(e + f*x)^(m + n + 2) /.
     {a -> 2, b -> -2, c -> -2, d -> 4, e -> 7, f -> -4, m -> 1, n -> 1/2, x -> 2}

evalues to $-2 i \sqrt{6}$. A correct antiderivative of $\int \frac{(a+b x)^m (c+d x)^n}{(e+f x)^{m+n+2}} \, dx$ is

$$\frac{(a+b x)^{m+1} (c+d x)^n \, _2F_1\left(m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right)}{(m+1) (b e-a f) (e+f x)^{m+n+1} \left(\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right)^n} $$ Albert