Fourier transform of a Bessel function multiplied by a Gaussian

An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $I_{mn}(\beta,q) = \int_{-\infty}^{\infty} e^{-\beta^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x) \,d x,$ where $m$ and $n$ are non-negative integers, and $J_{n+1/2}(\cdot)$ is a Bessel function of order $n+1/2$.