  Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring
of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and
let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of
characteristic $p > 0$ then let $I^*$ denote the tight closure of $I$. Let
$G_I(A)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$
with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show
that either the function $P_{\overline{I}} :\,n\mapsto
\lambda(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or
$\overline{I^n}=I^n$ for all $n\geq 1.$ We prove an analogus result for the
tight closure filtration if $A$ is of characteristic $p > 0$. When $A$ is
generalized Cohen-Macaulay and $I$ is generated by standard system of
parameters we give bounds for the first Hilbert coefficients of the integral
closure filtration of $I$ and the tight closure filtration of $I$.
