  We present the computation of a full set of planar five-point two-loop master
integrals with one external mass. These integrals are an important ingredient
for two-loop scattering amplitudes for two-jet-associated W-boson production at
leading color in QCD. We provide a set of pure integrals together with
differential equations in canonical form. We obtain analytic differential
equations efficiently from numerical samples over finite fields, fitting an
ansatz built from symbol letters. The symbol alphabet itself is constructed
from cut differential equations and we find that it can be written in a
remarkably compact form. We comment on the analytic properties of the integrals
and confirm the extended Steinmann relations, which govern the double
discontinuities of Feynman integrals, to all orders in $\epsilon$. We solve the
differential equations in terms of generalized power series on single-parameter
contours in the space of Mandelstam invariants. This form of the solution
trivializes the analytic continuation and the integrals can be evaluated in all
kinematic regions with arbitrary numerical precision.
