The disjunctive skolem chase is a sound and complete (albeit non-terminating) algorithm that can be used to solve conjunctive query answering over DL ontologies and programs with disjunctive existential rules. Even though acyclicity notions can be used to ensure chase termination for a large subset of real-world knowledge bases, the complexity of reasoning over acyclic theories still remains high. Hence, we study several restrictions which not only guarantee chase termination but also ensure polynomiality. We include an evaluation that shows that almost all acyclic DL ontologies do indeed satisfy these general restrictions.