  In previous work, the two-loop five-point amplitudes in $\mathcal{N}=4$ super
Yang-Mills theory and $\mathcal{N}=8$ supergravity were computed at symbol
level. In this paper, we compute the full functional form. The amplitudes are
assembled and simplified using the analytic expressions of the two-loop
pentagon integrals in the physical scattering region. We provide the explicit
functional expressions, and a numerical reference point in the scattering
region. We then calculate the multi-Regge limit of both amplitudes. The result
is written in terms of an explicit transcendental function basis. For certain
non-planar colour structures of the $\mathcal{N}=4$ super Yang-Mills amplitude,
we perform an independent calculation based on the BFKL effective theory. We
find perfect agreement. We comment on the analytic properties of the
amplitudes.
