  We obtain generalised power series expansions for a family of planar two-loop
master integrals relevant for the QCD corrections to Higgs + jet production,
with physical heavy-quark mass dependence. This is achieved by defining
differential equations along contours connecting two fixed points, and by
solving them in terms of one-dimensional generalised power series. The
procedure is efficient and can be repeated in order to reach any point of the
kinematic regions. The analytic continuation of the series is straightforward
and we present new results below and above the physical thresholds. The method
we use allows to compute the integrals in all kinematic regions with high
precision. Performing a series expansion on a typical contour above the
physical threshold takes on average $\mathcal{O}(1 \text{ second})$ per
integral with worst relative error of $\mathcal{O}(10^{-32})$, on a single CPU
core. After the series is found the numerical evaluation of the integrals in
any point of the contour is virtually instant. Our approach is general and can
be applied to Feynman integrals provided that a set of differential equations
is available.
