  In this work, we investigate the associated production of prompt heavy
quarkonium with a massive (anti)bottom quark to leading order in the NRQCD
factorization formalism at the LHC. We present numerical results for the
processes involving $J/\psi,\chi_{cJ},\Upsilon$ and $\chi_{bJ}$. From our work,
we find that the production rates of these processes are quite large, and these
processes have the potential to be detected at the LHC. When $p_T$ is smaller
than about 10 GeV, the $c\bar{c} [ ^1S_0^{(8)} ]$ state give the main
contribution to the $p_T$ distribution of prompt $J/\psi$ with a (anti)bottom
quark production. For the process of $pp \to \Upsilon+b(\bar{b})$, the
contribution of the CSM is larger than that in the COM at low $p_T$ region. We
also investigate the processes of $pp\to \chi_{cJ}+b(\bar{b})$ and $pp \to
\chi_{bJ}+b(\bar{b})$, in these processes, the $p_T$ distribution are dominated
by the CO Fock state contribution at the large $p_T$ region. These processes
provide an interesting signature that could be studied at the LHC, and the
measurement of these processes is useful to test the CSM and COM.
