We present a measurement of the total WW and WZ production cross sections in pp¯ collision at s=1.96 TeV, in a final state consistent with leptonic W boson decay and jets originating from heavy-flavor quarks from either a W or a Z boson decay. This analysis uses the full data set collected with the CDF II detector during Run II of the Tevatron collider, corresponding to an integrated luminosity of 9.4 fb-1. An analysis of the dijet mass spectrum provides 3.7σ evidence of the summed production processes of either WW or WZ bosons with a measured total cross section of σWW+WZ=13.7±3.9 pb. Independent measurements of the WW and WZ production cross sections are allowed by the different heavy-flavor decay patterns of the W and Z bosons and by the analysis of secondary-decay vertices reconstructed within heavy-flavor jets. The productions of WW and of WZ dibosons are independently seen with significances of 2.9σ and 2.1σ, respectively, with total cross sections of σWW=9.4±4.2 pb and σWZ=3.7-2.2+2.5 pb. The measurements are consistent with standard-model predictions.