The problem: -The integral of Bessel functions of different kind are not well placed among the literature. There is no clarification of how this integral works: \begin{equation} \int_{0}^{r}Y_{m}(\alpha_{1}z)J_{m}(\alpha_{2}z)z dz \end{equation} I arrived at this integral working on a particular solution for a quasi-linear wave operator during my research.
Obs: I tried to find something related on the works of Watson, Lebedev, Korenev and Gray et al but to no avail...
From DLMF, when $\alpha_1\ne\alpha_2$, \begin{equation} \int z Y_{m}\left(\alpha_1z\right)J_{m}(\alpha_2z)\mathrm{d}z=\frac{z% \left(\alpha_1Y_{m+1}\left(\alpha_1z\right)J_{m}(\alpha_2z)-\alpha_2Y_{% m}\left(\alpha_1z\right)J_{m+1}(\alpha_2z)\right)}{\alpha_1^{2}-\alpha_2^{2}} \end{equation} The value of this antiderivative is non zero at $z=0$. Indeed, for $z\to 0$, retaining the leading term \begin{align} J_n(z)&\sim \frac{1}{n!}\left( \frac{z}{2} \right)^n\\ Y_n(z)&\sim-\frac{(n-1)!}{\pi}\left( \frac{z}{2} \right)^{-n}\text{ for } n>0 \end{align} It exists thus a non-zero contribution of the term $zY_{m+1}\left(\alpha_1z\right)J_{m}(\alpha_2z)$: \begin{equation} \lim_{z\to 0} z\alpha_1Y_{m+1}\left(\alpha_1z\right)J_{m}(\alpha_2z)=-\frac{2}{\pi}\left( \frac{\alpha_2}{\alpha_1} \right)^m \end{equation} Then, \begin{align} \int_0^r z Y_{m}&\left(\alpha_1z\right)J_{m}(\alpha_2z)\mathrm{d}z=\\ &\frac{r% \left(\alpha_1Y_{m+1}\left(\alpha_1r\right)J_{m}(\alpha_2r)-\alpha_2Y_{% m}\left(\alpha_1r\right)J_{m+1}(\alpha_2r)\right)}{\alpha_1^{2}-\alpha_2^{2}}+\frac{2}{\pi}\left( \frac{\alpha_2}{\alpha_1} \right)^m\frac{1}{\alpha_1^{2}-\alpha_2^{2}} \end{align}