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This paper analyzes delay-dependent robust stability and H∞ performance of linear systems with an interval time-varying delay, based on a new Lyapunov–Krasovskii functional containing new triple integral terms. The time derivative of the Lyapunov–Krasovskii functional produces not only the strictly proper rational functions but also the non-strictly proper rational functions of the time-varying delays with first-order denominators. The combinations of the rational functions are directly handled via the Jensen inequality lemma and the lower bound lemma for reciprocal convexity, whereas such combinations were approximated in the literature. The proposed criteria become less conservative with the significantly smaller number of decision variables than the existing criteria, which will be demonstrated by some numerical examples.