  In this paper, the three-dimensional chemotaxis-stokes system
\begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla
n=\Delta n^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0, \medskip
c_t+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,\ \ t>0, \medskip u_t+\nabla
P=\Delta u +n\nabla\phi,&x\in\Omega,\ \ t>0, \nabla\cdot u=0, &x\in\Omega,\ \
t>0,, \end{array}\right. \end{eqnarray*} posed in a bounded domain
$\Omega\subset\mathbb{R}^3$ with smooth boundary is considered under the
no-flux boundary condition for $n$, $c$ and the Dirichlect boundary condition
for $u$ under the assumption that the Frobenius norm of the tensor-valued
chemotactic sensitivity $S(x,n,c)$ satisfies $S(x,n,c)<n^{l-2}\widetilde{S}(c)$
with $l>2$ for some non-decreasing function $\widetilde{S}\in
C^{2}((0,\infty))$. In present work, it is shown that the weak solution is
global in time and bounded while $m>m^\star(l)$, where \begin{eqnarray*}
m^\star(l)= \left\{\begin{array}{lll} \medskip l-\frac{5}{6},\ &\mathrm{if}\
\frac{31}{12}\geq l>2, \medskip \frac{7}{5}l-\frac{28}{15},\ &\mathrm{if}\
l>\frac{31}{12}. \end{array}\right. \end{eqnarray*}
