help with exponential integral and square root

Question

can somebody help me integrating this:

where m, p and q are positive constants. I tired change of variables and searched for the solution but could not find it.

Thanks

Note: this is the result of mathematica (zeta==a and rho==p):

but the Gamma(-0.25-0.5m) is illogical cuz this is infinity (gamma of a negative number) !!