  We generalize a Brian\c{c}on-Skoda type theorem first studied by Aberbach and
Huneke. With some conditions on a regular local ring $(R,\m)$ containing a
field, and an ideal $I$ of $R$ with analytic spread $\ell$ and a minimal
reduction $J$, we prove that for all $w \geq -1$, $ \bar{I^{\ell+w}} \subseteq
J^{w+1} \mathfrak{a} (I,J),$ where $\mathfrak{a}(I,J)$ is the coefficient ideal
of $I$ relative to $J$, i.e. the largest ideal $\mathfrak{b}$ such that
$I\mathfrak{b}=J\mathfrak{b}$. Previously, this result was known only for
$\m$-primary ideals.
