The static and dynamic behaviors of linear and ring polymers under shear flow over a wide range of shear rates are studied using a hybrid simulation method that couples multiple-particle collision dynamics with molecular dynamics. The results reveal that the polymer size increases monotonically with increasing shear rate when hydrodynamic interactions are ignored, in agreement with classic theoretical predictions. However, for the cases with hydrodynamic interactions, due to the transition from a linear to a nonlinear velocity profile, the size of both linear and ring chains exhibits a nonmonotonic dependence on the shear rate, and this counterintuitive behavior could be divided into three main regimes. At specific shear rates, linear polymers exhibit a relatively stable stretched state and a rapidly rotating collapsed state, which correspond to the maximum and minimum sizes, respectively. Although the rings behave similarly to the linear polymers, there exist two different relatively stable stretched states: one with an oval-shaped conformation and the other with an S-shaped conformation. For the oval-shaped state, the tumbling motion almost disappears but the tank-treading motion persists, whereas for the S-shaped state, both tumbling and tank-treading motions are greatly suppressed. Moreover, contrary to previous theoretical predictions, a noticeable bulge is observed for the polymer size in the gradient direction and the alignment angle deviates considerably from theoretical prediction, indicating the existence of relatively stable collapsed conformations at large shear rates for both linear and ring polymers. However, in ideal linear shear flow introduced by using the "fix deform" command to Dissipative Particle Dynamics system in LAMMPS package, both linear and ring polymers exhibit a monotonic dependence of size on shear rate. These results shed new light on the understanding of the dynamic response of linear and ring polymers in ultrahigh shear flows.