  We report that, unlike most inclusive quarkonium-production processes, the
production rate for p p -> J/psi+Upsilon+X dominantly depends on two not
well-known long-distance nonrelativistic QCD matrix elements,
<O_8^{J/psi}(^3S_1)> and <O_8^{Upsilon}(^3S_1)> at leading order in the strong
coupling constant, which account for the transition probabilities of the
color-octet spin-triplet heavy-quark-antiquark pairs c\bar{c}_8(^3S_1) -> J/psi
and b\bar{b}_8(^3S_1) -> Upsilon, respectively. With the integrated luminosity
~ 100 fb^{-1} at the center-of-momentum energy sqrt{s}=14 TeV we expect that
approximately 1900 p p -> J/psi+Upsilon+X events can be observed at the CERN
Large Hadron Collider by tagging muon pairs, which are enough to probe to the
color-octet mechanism. The forthcoming measurement may provide a useful
constraint to resolve the decade-old puzzle for the polarization of prompt
J/psi at the Fermilab Tevatron. If corresponding measured rate is significantly
less than the prediction, it may imply that the current values for the
color-octet matrix elements are overestimated.
