  Space-based gravitational-wave detectors such as LISA are expected to detect
inspirals of stellar-mass compact objects into massive black holes. Modeling
such inspirals requires fully relativistic computations to achieve sufficient
accuracy at leading order. However, subleading corrections such as the effects
of the spin of the inspiraling compact object may potentially be treated in
weak-field expansions such as the post-Newtonian (PN) approach.
  In this work, we calculate the PN expansion of eccentric orbits of spinning
bodies around Schwarzschild black holes. Then we use the Teukolsky equation to
compute the energy and angular momentum fluxes from these orbits up to the 5PN
order. Some of these PN orders are exact in eccentricity, while others are
expanded up to the tenth power in eccentricity. Then we use the fluxes to
construct a hybrid inspiral model, where the leading part of the fluxes is
calculated numerically in the fully relativistic regime, while the part linear
in the small spin is analytically approximated using the PN series. We
calculate LISA-relevant inspirals and respective waveforms with this model and
a fully relativistic model. Through the calculation of mismatch between the
waveforms from both models we conclude that the PN approximation of the
linear-in-spin part of the fluxes is sufficient for lower eccentricities.
