Here's some additional information:
I've recently encountered a very similar integral in my own work, with a ratio of gamma functions and a hypergeometric function and a Bessel function. The orders of both the hypergeometric function and the Bessel function depended on $s$. I was still able to produce a series solution consisting of terms containing the 1F1 and Bessel functions evaluated at the pole locations, since neither function had any poles or branch points in the $s$ plane.
I believe the same is true in your case, since it is the numerator parameter that depends upon $s$, and the 1F1 has no poles for this parameter. (Look at, for example, the functions.wolfram.com site for info on 1F1.)
Sorry for the overly introductory material in my earlier post, I was not sure of your familiarity with this method! You obviously know it well..
Hope this helps some,
Did this integral come from using Mellin transforms + Parseval to perform an integration? A standard thing to do now is to look at the large $t$ asymptotics of the integrand (where $s = \sigma + i t$) to determine if the contour can be moved across the poles of the gamma functions in the numerator, and in which direction. The constant $c$ would lie in the joint strip of analyticity of the Mellin transforms.
You then close the contour in the appropriate half plane and use the residue theorem to give either an asymptotic expansion or power series in the variable $b$. Sometimes these can be identified with known functions. This process all depends on the details of your parameters. Your situation is more complex because you have the 1F1(). Look up "Mellin-Barnes integrals" and you should find helpful information.