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Cosmin Crucean derived an analytical solution for the integral of a product of four Bessel functions and a power,

$$\int_{0}^{\infty} d x x^{\mu} J_{\alpha}(a x) J_{\beta}(a x) J_{\gamma}(b x) J_{\delta}(b x)$$

Similarly, in the Bateman Project Vol. II (one of the standard references for Bessel functions) on page 49 they show the integral for a product of two Bessel functions, a power and a Gaussian as shown in the screenshot:

p 49 of the Bateman Project vol. II

I'm looking for a solution to

$$\int_{0}^{\infty} d x x^{\mu} e^{-\zeta^2 x^2} J_{\alpha}(a x) J_{\beta}(b x) J_{\gamma}(c x) J_{\delta}(d x)$$

for which I haven't found any literature (and hence am not sure whether a closed form solution exists).

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