calculus - Integrals of the form $int x^{nu - 1}J_{nu}(x) mathrm{d}x$

Consider the following integral:

$$\int x^{\nu - 1}J_{\nu}(x) \ \mathrm{d}x,$$

where $\nu > 0$ and $J_{\nu}$ denotes the Bessel function of the first kind. For which values $\nu$ (either integers or half-integers) do we have a closed-form expression for this integral? For instance, in the case where $\nu = 1$, then we have the well known identity

$$\displaystyle \int J_{1}(x) \ \mathrm{d}x = -J_{0}(x).$$

If $\nu = 2$ then we also have

$$\displaystyle \int xJ_{2}(x) \ \mathrm{d}x = -2J_0(x) - xJ_1(x),$$

and it seems that for positive integers $\nu$ we can obtain an expression involving Bessel functions. Is there a general formula?

If $\nu$ is a half-integer then I am not sure a closed form expression exists. Wolfram Alpha says that

$$\displaystyle \int x^{1/2}J_{3/2}(x) \ \mathrm{d}x = \sqrt{\frac{2}{\pi}}\left(\mathrm{Si}(x) - \sin(x)\right),$$

where $\mathrm{Si}(x)$ denotes the sine integral. The presence of the $\sin$ also seems to suggest that the integral wouldn't converge on an infinite interval such as $[0, \infty)$, say.