The problem: -The solution of the integral of Bessel functions of first kind, but different order and argument is not well placed among the literature. There is no clarification of how this general integral works and if there is a closed form solution:
\begin{equation} \int_{0}^{r}J_{m}(\alpha_{1}z)J_{n}(\alpha_{2}z)z \text{d}z \end{equation}
in particular, the case of $m>0$ and $n=0$. While $\alpha_{1}$ and $\alpha_{2}$ are allowed to be complex or real.
Obs: I tried to find something related on the works of Watson, Lebedev, Korenev and Gray et al but to no avail...