What makes you believe that this function has an antiderivative that can be expressed in terms of not only elementary functions (in the technical sense of the term) of the various non-elementary functions that Mathematica knows about?
Would numerical integration over some prescribed interval of values of r
suffice for your purposes? If so, then for specific number a
and b
you want to use
NIntegrate[A B, {r, a, b}]