In this article, we present an effective approach to calculate quantum chemical two-electron integrals over basis sets consisting of Gaussian-type basis functions on graphical processing unit (GPU). Our framework generates several different variants called routes to the same integral problem with different integral algorithms (McMurchie–Davidson, Head-Gordon–Pople, and Rys) and precision. Each route is benchmarked on more GPU architectures, and with this data, a model is fitted to select the best available route for an integral task given a GPU architecture. Moreover, this approach supports the computation of high angular momentum orbitals up to g effectively on GPU, tested up to cc-pVQZ-sized basis sets. Rigorous analysis is shown regarding the effectiveness of our method. Molecule simulations with several basis sets are measured using NVIDIA GTX 1080 Ti, NVIDIA P100, and NVIDIA V100 cards.