  In this paper we establish the generalized Beukers integral
$I_{m}(a_{1},...,a_{n})$ with some methods of partial fraction decomposition.
Thus one obtains an explicit expression of the generalized Beukers integral.
Further, we estimate the rational denominator of $I$ and. In the second section
of this paper, we provide some estimates of the upper and lower bound of the
value $J_{3}$, which involves the generalized Beukers integral and is related
to $\zeta(5)$.
