This paper addresses the delay-dependent stability for systems with time-varying delay. First, by taking multi-integral terms into consideration, new Lyapunov-Krasovskii functional is defined. Second, in order to reduce the computational complexity of the main results, reciprocally convex approach and some special transformations are introduced, and new delay-dependent stability criteria are proposed, which are less conservative and have less decision variables than some previous results. Finally, two well-known examples are given to illustrate the correctness and advantage of our theoretical results.