Pentagon integrals to arbitrary order in the dimensional regulator

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