Thanks for that. My immediate interest is actually for a=b, so I am probably stuck trying to do the integral (or letting mathematica have a crack). I was hoping somebody may know a reference where this is written down.
I just looked at the density (eqn (1) in
Schroder).
With that, it looks like your general case is indeed expressible in terms of the
confluent hypergeometric function,
as Mathematica can do the general integral:
Integrate[w^p E^(-w) BesselI[k,2 Sqrt[x w]],{w,0,Infinity}]
= x^(k/2) Gamma[1+k/2+p] Hypergeometric1F1Regularized[1+k/2+p,1+k,x]
subject to some conditions on the parameters: p > -1 && (k/2)+p > -1.
Also Hypergeometric1F1Regularized(a,b,z) = [$] M(a,b,z)/\Gamma(b)[$] in more standard notation.
So, clean it up and you've got your answer.