Integral over product of two bessel functions and power

I have searched the literature on integrals over bessel functions, but I couldn't find anything. The integral to be evaluated is,

$\int_0^a J_{n}(bx)J_{\mu}(cx)xdx =: \mathcal{M}_r(a;n,\mu;b,c)$

where $J_n(x)$ is a Bessel function of integer order, $J_{\mu}(x)$ is a Bessel function of order $µ$, $a,b,c\in\mathbb{R}^+$. The RHS object denotes another expression having the arguments inscribed that solves the integral as good as possible.

A colleage couldn't get mathematica to yield a result, and I couldn't evaluate the integral using tables etc.

This looks different from what you exactly need, but with a simple transformation, you can find solution for when $b=c$. Further, if you deeply go through the literature, you can also find more general solution when $b\neq c$. Further, In your case $\lambda=1$: $$\int_0^a x^{\lambda } J_{\mu }(x) J_{\nu }(x) \, dx=\frac{a^{\lambda +\mu +\nu +1} }{2^{\mu +\nu } \Gamma (\mu +1) \Gamma (\nu +1) (\lambda +\mu +\nu +1)}\times\, _3F_4\left(\frac{1}{2} (\mu +\nu +1),\frac{1}{2} (\mu +\nu +2),\frac{1}{2} (\lambda +\mu +\nu +1);\mu +1,\nu +1,\mu +\nu +1,\frac{1}{2} (\lambda +\mu +\nu +3);-a^2\right)$$ which includes hypergeometric function.