  We present the first unquenched lattice-QCD calculation of the form factors
for the decay $B\rightarrow D^\ast\ell\nu$ at nonzero recoil. Our analysis
includes 15 MILC ensembles with $N_f=2+1$ flavors of asqtad sea quarks, with a
strange quark mass close to its physical mass. The lattice spacings range from
$a\approx 0.15$ fm down to $0.045$ fm, while the ratio between the light- and
the strange-quark masses ranges from 0.05 to 0.4. The valence $b$ and $c$
quarks are treated using the Wilson-clover action with the Fermilab
interpretation, whereas the light sector employs asqtad staggered fermions. We
extrapolate our results to the physical point in the continuum limit using
rooted staggered heavy-light meson chiral perturbation theory. Then we apply a
model-independent parametrization to extend the form factors to the full
kinematic range. With this parametrization we perform a joint
lattice-QCD/experiment fit using several experimental datasets to determine the
CKM matrix element $|V_{cb}|$. We obtain $\left|V_{cb}\right| = (38.40 \pm
0.68_{\textrm{th}} \pm 0.34_{\textrm{exp}} \pm 0.18_{\textrm{EM}})\times
10^{-3}$. The first error is theoretical, the second comes from experiment and
the last one includes electromagnetic and electroweak uncertainties, with an
overall $\chi^2\text{/dof} = 126/84$, which illustrates the tensions between
the experimental data sets, and between theory and experiment. This result is
in agreement with previous exclusive determinations, but the tension with the
inclusive determination remains. Finally, we integrate the differential decay
rate obtained solely from lattice data to predict $R(D^\ast) = 0.265 \pm
0.013$, which confirms the current tension between theory and experiment.
