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.



Definite integral of Bessel function of the first kind times $x^{3/2}$
The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function:
$$\int\_0^a x^{3/2} J\_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\, \_1F\_2\left(\frac{\nu}{2}+\frac{5}{4};\frac{\nu}{2}+\frac{9}{4},\nu+1;-\frac{1}{4} a^2 b^2\right).$$


