The integral is $$\int_k^\infty x^{-b} J_0(x) e^{-a x^2} dx$$ where $k>0$ is some constant, $b$ is rational, and $a>0$ some constant.
I have seen in the Gradshtein I. S., Ryzhik I. M. Tables Of Integrals, Series And Products book (equation 6.631 (1) ) a similar integral from zero to infinity, but my specific integral does not converge in this range since the integrand blows up at zero, and hence I must do it from some positive constant k. The equation in that textbook was solved by expressing the Bessel function as its series and then integrating term by term as described by Herbert Buchholz in the book The Confluent Hypergeometric Function.
To follow this line of reasoning, I have tried writing the Bessel function in terms of its series and integrating term by term for my integral, but I am left with the series $$\sum_{m=0}^{\infty}\frac{(-1)^m a^{-(2m-b+1)/2}}{m!m! 2^{2m+1}}\Gamma((2m-b+1)/2, a k^2) $$ which I dont know how to express in closed form.
Any help would be appreciated.