Sample pooling has the potential to significantly enhance large-scale screening procedures, especially in scenarios like the COVID-19 pandemic, where rapid and widespread PCR testing has been crucial. Efficient strategies are essential to increase the testing capacity, i.e., the number of tests that can be processed within a given timeframe. Nonadaptive pooling strategies can further streamline the testing process by reducing the required testing rounds. In contrast to adaptive strategies, where subsequent tests depend on prior results, nonadaptive pooling processes all sample in a single round, eliminating the need for sequential retesting and reducing delays. This paper presents a highly flexible method based on integer programming to design optimized pooling patterns suitable for various applications, including medical diagnostics and quality control in industrial production. Using coronavirus testing as a case study, we formulate and solve optimization and satisfiability models that compute efficient pool designs. Our optimized pooling not only does increase testing capacity but also accelerates the testing process and reduces overall costs. The proposed method is adaptable and can be seamlessly integrated into automated testing systems.