We present an analysis of the semi-inclusive decays $B_s to D_s^- X ell^+ u$ and $B_s to D_s^{*-} X ell^+ u$, where $X$ denotes a final state that may consist of additional hadrons or photons and $ell$ is an electron or muon. The studied $B_s$ decays are contained in the $121.4~{rm fb}^{-1}$ $Upsilon(5S)$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. The branching fractions of the decays are measured to be $mathcal{B}(B_s to D_s^- X ell^+ u) = [8.2 pm 0.2 (text{stat}) pm 0.6 (text{syst}) pm 1.4 (text{ext})]%$ and $mathcal{B}(B_s to D_s^{*-} X ell^+ u) = [5.4 pm 0.4 (text{stat}) pm 0.4 (text{syst}) pm 0.9 (text{ext})]%$, where the first two uncertainties are statistical and systematic and the last is due to external parameters. The measurement also provides an estimate of the $B_s^{(*)}bar{B}_s^{(*)}$ production cross-section, $sigma(e^+e^- to B_s^{(*)}bar{B}_s^{(*)}) = (53.8 pm 1.4 (text{stat}) pm 4.0 (text{syst}) pm 3.4 (text{ext})),{rm pb}$, at the center-of-mass energy $sqrt{s} = 10.86,{rm GeV}$.
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