  The progress in optical clock with uncertainty at a level of $10^{-18}$
requires unprecedented precision in estimating the contribution of multipolar
and higher-order effects of atom-field interactions. Current theoretical and
experimental results of dynamic multipolar polarizabilities and
hyperpolarizabilities at the magic wavelength for the Sr clock differ
substantially. We develop a combined approach of the Dirac-Fock plus core
polarization (DFCP) and relativistic configuration interaction (RCI) methods to
calculate dynamic multipolar polarizabilities and hyperpolarizabilities of the
Sr atom. Our differential dynamic hyperpolarizability at the magic wavelength
is $-2.09(43)\times10^{7}$ a.u., which is consistent with the existing
theoretical and experimental results. Our differential multipolar
polarizability is $2.68(94)\times 10^{-5}$ a.u., which validates independently
the theoretical work of Porsev {\em et al.} [Phys. Rev. Lett. 120, 063204
(2018)], but different from recent measurement of Ushijima {\em et al.} [Phys.
Rev. Lett. 121, 263202 (2018)].
