I would like to solve this integral: \begin{equation} \int_0^1 J_d(\alpha x) J_c(\beta x) J_b( \gamma x) J_a(\rho x) x dx \end{equation}
Where a, b, c, d are positive integers (including 0), and $\alpha$, $\beta$, $\gamma$, $\rho$ are positive real numbers.
In fact, my real problem is to find the average in space between the multiplication of any 4 modes accepted in a multi-mode fiber: \begin{equation} E[ M_{nm} M_{n'm'} M_{n''m''} M_{n'''m'''}] \end{equation}
Where each mode is defined in polar coordinates: \begin{equation} M_{nm} = J_m(u_n r /R) \cos( m\phi ) \end{equation}
where $\phi \in [0, 2\pi ]$ and $r \in [0, R ]$. Hence, the calculation of $E[ M_{nm} M_{n'm'} M_{n''m''} M_{n'''m'''}]$ can be separated into the radial integral and the angular integral. I already solved the angular one. Which tells me that $E[ M_{nm} M_{n'm'} M_{n''m''} M_{n'''m'''}]$ is not zero only if $m \pm m' \pm m'' \pm m''' = 0$. In this cases, that result is $\frac{\pi}{4}$. So I have the radial part left to solve, which is the one I state in this question.
It turn, I need to calculate $E[ M_{nm} M_{n'm'} M_{n''m''} M_{n'''m'''}]$ to calculate the variance of a light intensity pattern measured at the end facet of the multi-mode fiber.