I am looking for preferably a closed form (or series solution if not possible) for the following integral:

$$\int_0^a x^{3/2} J_\nu (bx) dx$$

where $\nu$ is an integer. This 1D integral appears when taking the polar Fourier transform of a separable radially symmetric function in 2D that I would like to propagate using the angular spectrum method. I understand that I can express this as a finite Hankel transform of $\sqrt{x}$ but I was hoping there was an analytic solution for this simple case.