© 2016 American Physical Society. We describe a measurement of the ratio of the cross sections times branching fractions of the Bc+ meson in the decay mode Bc+→J/ψμ+ν to the B+ meson in the decay mode B+→J/ψK+ in proton-antiproton collisions at center-of-mass energy 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 fb-1. The ratio of the production cross sections times branching fractions for Bc+ and B+ mesons with momentum transverse to the beam greater than 6 GeV/c and rapidity magnitude smaller than 0.6 is 0.211±0.012(stat)-0.020+0.021(syst). Using the known B+→J/ψK+ branching fraction, the known B+ production cross section, and a selection of the predicted Bc+→J/ψμ+ν branching fractions, the range for the total Bc+ production cross section is estimated.