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We investigate the asymptotic expansion of integrals analogous to Ball'sintegral \[\int_0^\infty\left(\frac{\Gamma(1+\nu)|J_\nu(x)|}{(x/2)^\nu}\right)^{\!n}dx\] for large $n$in which the Bessel function $J_\nu(x)$ is replaced by the modified Besselfunctions $I_\nu(x)$ and $K_\nu(x)$ together with appropriate exponentialfactors $e^{\mp x}$, respectively.  The above integral with $J_\nu(x)$ replaced by a hyper-Bessel function of thetype recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] andtaken over a finite interval determined by the first positive zero of thefunction is also considered for $n\to\infty$. We give the leading asymptoticbehaviour of the hyper-Bessel function for $x\to+\infty$ in an appendix.Numerical examples are given to illustrate the accuracy of the variousexpansions obtained.

Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions