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This paper is concerned with robust stability of uncertain linear systems with interval time-varying delay. The time-varying delay is assumed to belong to an interval and the derivative of the time-varying delay has upper bound, at the same time, no restriction on the derivative of the time-varying delay is considered, which allows the delay to be a fast time-varying function. The uncertainty under consideration includes a polytopic-type uncertainty and a linear fractional norm-bounded uncertainty. In order to obtain much less conservative results, a new Lyapunov-Krasovshii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is proposed to derive some new stability criteria. Moreover, no redundant matrix variable is introduced. Finally, numerical examples are given to demonstrate the effectiveness of the proposed stability criteria.