  $\Upsilon(nS)$ and $\chi_b(nP)$ (n=1,2,3) production at the LHC is studied at
next-to-leading order in $\alpha_s$ in nonrelativistic QCD. Feeddown
contributions from higher $\chi_b$ and $\Upsilon$ states are all considered for
lower $\Upsilon$ cross sections and polarizations. The long distance matrix
elements (LDMEs) are extracted from the yield data, and then used to make
predictions for the $\Upsilon(nS)$ polarizations, which are found to be
consistent with the measured polarization data within errors. In particular,
the $\Upsilon(3S)$ polarization puzzle can be understood by a large feeddown
contribution from $\chi_b(3P)$ states. Our results may provide a good
description for both cross sections and polarizations of prompt $\Upsilon(nS)$
and $\chi_b(nP)$ production at the LHC.
