  We analytically calculate one-loop five-point Master Integrals,
\textit{pentagon integrals}, with up to one off-shell leg to arbitrary order in
the dimensional regulator in $d=4-2\epsilon$ space-time dimensions. A pure
basis of Master Integrals is constructed for the pentagon family with one
off-shell leg, satisfying a single-variable canonical differential equation in
the Simplified Differential Equations approach. The relevant boundary terms are
given in closed form, including a hypergeometric function which can be expanded
to arbitrary order in the dimensional regulator using the \texttt{Mathematica}
package \texttt{HypExp}. Thus one can obtain solutions of the canonical
differential equation in terms of Goncharov Polylogartihms of arbitrary
transcendental weight. As a special limit of the one-mass pentagon family, we
obtain a fully analytic result for the massless pentagon family in terms of
pure and universally transcendental functions. For both families we provide
explicit solutions in terms of Goncharov Polylogartihms up to weight four.
