How to integrate a binomial expression without expanding it before?

Let us generalise a bit. The integral of $(a+bx^n)^m$ is given by (just expanding and integrating term-wise):

$$\int(a+bx^n)^m\,\mathrm dx = \sum_{k=0}^m \binom m ka^{m-k}b^k\frac{x^{kn+1}}{kn+1}$$

Now I have not ever seen a nice form for a general expression like the sum on the right-hand side.

In fact, when I just look at it (for example when handed a similar sum in a different context), I usually think nothing except "Hey, that's the integral of $(a+bx^n)^m$".

For many usages of sums like the one on the right, the integral representation is actually a way of simplifying the problem; if there were nice closed forms currently (widely) known, the mathematical practice would probably en masse use those instead of the integral representation.

This all provides some circumstantial evidence to support the gut feeling that there probably is no general way.