 Caccioppoli inequality with bounded lower-order terms

For $\mathrm{div}(ADu)+b\cdot Du+cu=f$ with bounded measurable coefficients and uniformly elliptic $A$, testing with ${\eta }^{2}u$ and using Young inequality gives $\int {\eta }^2|Du{|}^2\le C\int (1+|D\eta {|}^2)u^2+C\int {\eta }^2{f}^2$. No sign condition on $c$ is needed for this interior estimate. A cutoff equal to one on $U$ and supported in $V$ yields the displayed bound; the constant depends only on dimension, ellipticity, coefficient bounds and the separation of $U$ from the boundary of $V$.