We present a measurement of the total {it WW} and {it WZ} production cross sections in $pbar{p}$ collision at $sqrt{s}=1.96$ 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 $3.7sigma$ evidence of the summed production processes of either {it WW} or {it WZ} bosons with a measured total cross section of $sigma_{WW+WZ} = 13.7pm 3.9$~pb. Independent measurements of the {it WW} and {it 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 {it WW} and of {it WZ} dibosons are independently seen with significances of $2.9sigma$ and $2.1sigma$, respectively, with total cross sections of $sigma_{WW}= 9.4pm 4.2$~pb and $sigma_{WZ}=3.7^{+2.5}_{-2.2}$~pb. The measurements are consistent with standard-model predictions.
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