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In this article, the issue of robust stability analysis for a sort of uncertain neutral system with mixed time-varying delays is studied. A new Lyapunov–Krasovskii functional comprising quadruple-integral term is introduced so as to develop a less conservative stability condition. A novel discrete and neutral delay-dependent stability criterion based on linear matrix inequalities is given using delay-central point method as well as reciprocally convex combination approach, which is derived by integral inequality approach. Compared with the existing literature, this criterion can greatly reduce the complexity of theoretical derivation and computation. Finally, three numerical comparative examples are designated to verify the superiority of the proposed approach in reducing the conservation of conclusion.