Order $1/m_b^3$ corrections to inclusive semileptonic $B$ decay

We compute the order $1/m_b^3$ nonperturbative contributions to the inclusive differential $B\rightarrow X_c\ell\bar\nu$ decay rate. They are parametrized by the expectation values of two local and four nonlocal dimension-six operators. We present the corrections to both the lepton and hadron spectra. The correction to the total rate turns out to be enhanced by $\log(m_c^2/m_b^2)$.