Because, like you said you can expand it by binomial to get a polynomial. And that polynomial can easily be expressed and its integral.
But it might be messy, I did not try.
Am I right to assume this is some problem you were given to find what the value is in terms of n and m? If thus, I might work on it if I have time.
The two recommendations I would say the is basic one, explain by binomial series.
The second is so use repeated use of integration by parts.