In this paper, we deal with the robust mixed $H_2/H_{infty}$ guaranteed-cost control problem involving uncertain neutral stochastic distributed delay systems. More precisely, the aim of this problem is to design a robust mixed $H_2/H_{infty}$ guaranteed-cost controller such that the close-loop system is stochastic mean-square exponentially stable, and an $H_2$ performance measure upper bound is guaranteed, for a prescribed $H_{infty}$ attenuation level ${gamma}$. Therefore, the fast convergence can be fulfilled and the proposed controller is more appealing in engineering practice. Based on the Lyapunov-Krasovskii functional theory, new delay-dependent sufficient criteria are proposed to guarantee the existence of a desired robust mixed $H_2/H_{infty}$ guaranteed cost controller, which are derived in terms of linear matrix inequalities(LMIs). Furthermore, the design problem of the optimal robust mixed $H_2/H_{infty}$ guaranteed cost controller, which minimized an $H_2$ performance measure upper bound, is transformed into a convex optimization problem with LMIs constraints. Finally, two simulation examples illustrate the design procedure and verify the expected control performance.