  The Brian\c{c}on-Skoda theorem in its many versions has been studied by
algebraists for several decades. In this paper, under some assumptions on an
F-rational local ring $(R,\m)$, and an ideal $I$ of $R$ of analytic spread
$\ell$ and height $g < \ell$, we improve on two theorems by Aberbach and
Huneke. Let $J$ be a reduction of $I$. We first give results on when the
integral closure of $I^\ell$ is contained in the product $J I_{\ell-1}$, where
$I_{\ell-1}$ is the intersection of the primary components of $I$ of height
$\leq \ell-1$. In the case that $R$ is also Gorenstein, we give results on when
the integral closure of $I^{\ell-1}$ is contained in $J$.
