This work addresses a challenging inventory routing problem that arises from a practical application faced by air-product companies, including Air Liquide. Given its computational complexity and industrial importance, this problem (denoted as IRP-Challenge2016) was presented as the topic of the French Operational Research and Decision Support Society/European Operational Research Society (ROADEF/EURO) Challenge 2016. The IRP-Challenge2016 seeks an optimal delivery schedule to minimize the unit distribution cost, while satisfying various hard constraints. It involves a single product, a heterogeneous fleet, heterogeneous drivers, multiperiods, a deterministic consumption forecast, and time-window constraints. We present a new mathematical formulation of the problem and introduce a matheuristic algorithm that integrates a local search-based metaheuristic with mathematical programming. Our algorithm combines mixed integer programming and linear programming as slave methods to optimize timing and delivery and embeds these procedures within a multineighborhood search metaheuristic to adjust routes. The method extends and enhances a preliminary version of our algorithm, which ranked third in the final round of the ROADEF/EURO Challenge 2016. Computational results for 20 challenge benchmark instances demonstrate the value of the proposed algorithm in terms of both effectiveness and efficiency with respect to the results reported in the competition. We additionally analyze several key components of our matheuristic to gain an insight into its operation.