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