  The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived
from two compact Coxeter polytops which are combinatorially simplicial prisms
(or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic
space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding
hyperbolic tilings are generated by reflections through their delimiting
hyperplanes those involve the study of the relating densest hyperball
(hypersphere) packings with congruent hyperballs.
  The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in the
hyperbolic spaces $\mathbf{H}^n$ $(n=3,4)$. In this paper we extend this
procedure to determine the optimal hyperball packings to the above
5-dimensional prism tilings. We compute their metric data and the densities of
their optimal hyperball packings, moreover, we formulate a conjecture for the
candidate of the densest hyperball packings in the 5-dimensional hyperbolic
space $\mathbf{H}^5$.
