Summary: Nonconvex derivative-free time-consuming objectives are often optimized using “black-box” optimization. These approaches assume very little about the objective. While broadly applicable, they typically require more evaluations than methods exploiting more problem structure. Often, such time-consuming objectives are actually the sum or integral of a larger number of functions, each of which consumes significant time when evaluated individually. This arises in designing aircraft, choosing parameters in ride-sharing dispatch systems, and tuning hyperparameters in deep neural networks. We develop a novel Bayesian optimization algorithm that leverages this structure to improve performance. Our algorithm is average-case optimal by construction when a single evaluation of the integrand remains within our evaluation budget. Achieving this one-step optimality requires solving a challenging value of information optimization problem, for which we provide a novel efficient discretization-free computational method. We also prove consistency for our method in both continuum and discrete finite domains for objective functions that are sums. In numerical experiments comparing against previous state-of-the-art methods, including those that also leverage sum or integral structure, our method performs as well or better across a wide range of problems and offers significant improvements when evaluations are noisy or the integrand varies smoothly in the integrated variables.