  Inclusive production of $\Upsilon J/\psi$ pair in proton-proton interation at
LHCb is considered. This process is forbidden at leading order of perturbation
theory, so such channels as double parton scattering, $\chi_b\chi_c$ pair
production with subsequent radiative decays of $P$-wave quarkonia,
contributions of color-octet states, and NLO corrections are studied in
details. For all these channels we present theoretical predictions of total
cross sections at LHCb and distributions over different kinematical variables.
According to presented in the paper results, double parton interaction gives
main contribution to the cross section of the considered reaction.
