  Let $f$ be a convergent power series of $n$ variables having an isolated
singularity at 0. For a rational number $\alpha$, setting $(X,0)=({\mathbb
C}^n,0)$, we show that the length of the ${\mathcal D}_X$-module ${\mathcal
D}_Xf^{-\alpha}$ is given by
$\widetilde{\nu}_{\alpha}+r_f\widetilde{\delta}_{\alpha}+1$. Here $r_f$ is the
number of local irreducible components of $f^{-1}(0)$ (with $r_f=1$ for $n>2$),
$\widetilde{\nu}_{\alpha}$ is the dimension of the graded piece ${\rm
Gr}_V^{\alpha}$ of the $V$-filtration on the saturation of the Brieskorn
lattice modulo the image of $N:=\partial_tt-\alpha$ on ${\rm Gr}_V^{\alpha}$ of
the Gauss-Manin system, and $\widetilde{\delta}_{\alpha}:=1$ if
$\alpha\in{\mathbb Z}_{>0}$, and 0 otherwise. This theorem can be proved also
by employing a generalization a recent formula of T. Bitoun in the integral
exponent case. The theorem generalizes an assertion by T. Bitoun and T.
Schedler in the weighted homogeneous case where the saturation coincides with
the Brieskorn lattice and $N=0$. In the semi-weighted-homogeneous case, our
theorem implies some sufficient conditions for their conjecture about the
length of ${\mathcal D}_Xf^{-1}$ to hold or to fail.
