  The non-first-order-factorizable contributions (The terms
'first-order-factorizable contributions' and 'non-first-order-factorizable
contributions' have been introduced and discussed in Refs.
\cite{Behring:2023rlq,Ablinger:2023ahe}. They describe the factorization
behaviour of the difference- or differential equations for a subset of master
integrals of a given problem.) to the unpolarized and polarized massive
operator matrix elements to three-loop order, $A_{Qg}^{(3)}$ and $\Delta
A_{Qg}^{(3)}$, are calculated in the single-mass case. For the $_2F_1$-related
master integrals of the problem, we use a semi-analytic method based on series
expansions and utilize the first-order differential equations for the master
integrals which does not need a special basis of the master integrals. Due to
the singularity structure of this basis a part of the integrals has to be
computed to $O(\varepsilon^5)$ in the dimensional parameter. The solutions have
to be matched at a series of thresholds and pseudo-thresholds in the region of
the Bjorken variable $x \in ]0,\infty[$ using highly precise series expansions
to obtain the imaginary part of the physical amplitude for $x \in ]0,1]$ at a
high relative accuracy. We compare the present results both with previous
analytic results, the results for fixed Mellin moments, and a prediction in the
small-$x$ region. We also derive expansions in the region of small and large
values of $x$. With this paper, all three-loop single-mass unpolarized and
polarized operator matrix elements are calculated.
