How to Evaluate Integrals Involving Bessel Functions and Exponential Terms?

[ibw]

I am doing a research degree in optical fields and ended up with the following integral in my math model. can you suggest any method to evaluate this integral please. Thanks in advance

∫(j(x) *e^(ax^2+ibx^2) dx


J --> zero order bessel function
i--. complex
a & b --> constants

 

[Thaakisfox]

If the limits are from 0 to infinity it can be easily evaluated using Webers double integral.
for other limits I don't think a close form exists.
Check: Watson: A treatise on the theory of Bessel functions
for more details