We describe a measurement of the ratio of the cross sections times branching fractions of the $B_c^+$ meson in the decay mode $B_c^+ rightarrow J/psi mu u$ to the $B^+$ meson in the decay mode $B^+ rightarrow J/psi K^+$ in proton-antiproton collisions at center-of-mass energy $sqrt{s}=1.96$ TeV. The measurement is based on the complete CDF Run II data set, which comes from an integrated luminosity of $8.7,{rm fb}^{-1}$. The ratio of the production cross sections times branching fractions for $B_c^+$ and $B_c^+$ mesons with momentum transverse to the beam greater than $6~textrm{GeV}/c$ and rapidity magnitude smaller than 0.6 is $0.211pm 0.012~mbox{(stat)}^{+0.021}_{-0.020}~mbox{(syst)}$. Using the known $B^+ rightarrow J/psi K^+$ branching fraction, the known $B^+$ production cross section, and a selection of the predicted $B_c^+ rightarrow J/psi mu u$ branching fractions, the range for the total $B_c^+$ production cross section is estimated.
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