This paper addresses the containment control of a multi-agent system with dynamic leaders while considering the state time-delay and time-varying input disturbances. The agents dynamics are assumed to be high-order integrals and an unkown state time-delay with a known upper bound is considered. In order to deal with dynamic leaders and time-varying input disturbances, an appropriate distributed PIp-type-based control protocol is proposed in this study. Thereafter, utilising the Lyapunov-Krasovskii stability theorem and based on LMI (Linear Matrix Inequality) approach, two theorems are proved which guarantee asymptotically stability of the multi-agent system with fixed and switching networks. In addition to stability assurance, the gains of the proposed protocol are achieved through solving the LMI. Finally, some numerical examples are given to demonstrate the validity of the theoretical results.