We present analytic expressions in terms of Goncharov Polylogarithms up to transcendental weight four, for pentagon integrals with massless propagators and with up to one external mass, in d = 4 − 2 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 Mathematica package HypExp. Thus one can obtain solutions of the canonical differential equation to arbitrary order in the dimensional regulator. The x → 1 limit can provide solutions for the massless pentagon family.