  We study the spectra of the bottomonium and B_c states within perturbative
QCD up to order alpha_s^4. The O(Lambda_QCD) renormalon cancellation between
the static potential and the pole mass is performed in the epsilon-expansion
scheme. We extend our previous analysis by including the (dominant) effects of
non-zero charm-quark mass in loops up to the next-to-leading non-vanishing
order epsilon^3. We fix the b-quark MSbar mass $\bar{m}_b \equiv m_b^{\bar{\rm
MS}}(m_b^{\bar{\rm MS}})$ on Upsilon(1S) and compute the higher levels. The
effect of the charm mass decreases $\bar{m}_b$ by about 11 MeV and increases
the n=2 and n=3 levels by about 70--100 MeV and 240--280 MeV, respectively. We
provide an extensive quantitative analysis. The size of non-perturbative and
higher order contributions is discussed by comparing the obtained predictions
with the experimental data. An agreement of the perturbative predictions and
the experimental data depends crucially on the precise value (inside the
present error) of alpha_s(M_Z). We obtain $m_b^{\bar{\rm MS}}(m_b^{\bar{\rm
MS}}) = 4190 \pm 20 \pm 25 \pm 3 ~ {\rm MeV}$.
