I'm searching for a way to evaluate the following integral: $$\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$$
where $J_v(x)$ are the Bessel-functions, and $v \in \mathbb{N}, (a,\beta) \in \mathbb{R}, \alpha \in \mathbb{C}$. The integral has a closed form when $a \to \infty$ (Gradshteyn/Ryzhik). However, is there also a way to solve the integral with a general upper interval-limit?