We examine the performance of various commonly used integration schemes in dissipative particle dynamics simulations. We consider this issue using three different model systems, which characterize a variety of different conditions often studied in simulations. Specifically, we clarify the performance of integration schemes in hybrid models, which combine microscopic and mesoscale descriptions of different particles using both soft and hard interactions. We find that in all three model systems many commonly used integrators may give rise to surprisingly pronounced artifacts in physical observables such as the radial distribution function, the compressibility, and the tracer diffusion coefficient. The artifacts are found to be strongest in systems, where interparticle interactions are soft and predominated by random and dissipative forces, while in systems governed by conservative interactions the artifacts are weaker. Our results suggest that the quality of any integration scheme employed is crucial in all cases where the role of random and dissipative forces is important, including hybrid models where the solvent is described in terms of soft potentials. Regarding the integration schemes, the best overall performance is found for integrators in which the velocity dependence of dissipative forces is taken into account, and particularly good performance is found for an approach in which velocities and dissipative forces are determined self-consistently. Remaining temperature deviations from the desired limit can be corrected by carrying out the self-consistent integration in conjunction with an auxiliary thermostat, in a manner that is similar in spirit to the well-known Nose-Hoover thermostat. Further, we show that conservative interactions can play a significant role in describing the transport properties of simple fluids, in contrast to approximations often made in deriving analytical theories. In general, our results illustrate the main problems associated with simulation methods in which dissipative forces are velocity dependent, and point to the need to develop new techniques to resolve these issues.