Yes, the problem is the expansion of the factor $$\left(\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right)^{-n}$$

in my proposed antiderivative to $$\left(\frac{b (c+d x)}{b c-a d}\right)^{-n} \left(\frac{b (e+f x)}{b e-a f}\right)^n$$ in Mathematica's result, which unfortunately is not always valid if n is not an integer.

Albert

Hi Albert,

I would be careful with the use of the term "incorrect". One has to make proper assumption to enable Mathematica to do cancelations. Try this:

ClearAll["Global`*"]
$Assumptions = {n >= 0, n \[Element] Integers};
origExpr = ((a + b x)^m (c + d x)^n)/(e + f x)^(m + n + 2);
iexpr = Integrate[origExpr, x];
dexpr = D[iexpr, x] // Simplify;
dexpr === origExpr
(* Out:  True *)

Regards -- Henrik

PS.: I hope this is not one of those cases you count as "Mathematica fails" on your website ...