If your variable of integration is y, and m and T are to remain symbolic (no definite value given for them before the integration), then you will have trouble finding an analytic integral. You effectively have a y^m term as part of f, and the integral of that part can look very different depending on m (especially m=0 or m=-1)
There is a transform you can do (in Maple at least) that converts the integral into an infinite sum. Not that it does you much good as you end up having to evaluate the infinite sum numerically. And the transform turns out not to be valid if m is integral.
Matters become much easier if you are given m before you do the integration.
If your m are strictly positive integer, then you might be able to work out the pattern. I can see that there is a pattern, but the rule for the the coefficients of the polynomials involved are not obvious.