Collision detection plays a key role in the simulation of interacting rigid bodies. However, owing to its computational complexity current methods typically prioritize either maximizing processing speed or fidelity to real-world behaviors. Fast real-time detection is achieved by simulating collisions with simple geometric shapes whereas incorporating more realistic geometries with multiple points of contact requires considerable computing power which slows down collision detection. In this work, we present a new approach to modeling and simulating collision-inclusive multibody dynamics by leveraging computer algebra system (CAS). This approach offers flexibility in modeling a diverse set of multibody systems applications ranging from human biomechanics to space manipulators with docking interfaces, since the geometric relationships between points and rigid bodies are handled in a generalizable manner. We also analyze the performance of integrating this symbolic modeling approach with collision detection formulated either as a traditional overlap test or as a convex optimization problem. We compare these two collision detection methods in different scenarios and collision resolution using a penalty-based method to simulate dynamics. This work demonstrates an effective simplification in solving collision dynamics problems using a symbolic approach, especially for the algorithm based on convex optimization, which is simpler to implement and, in complex collision scenarios, faster than the overlap test.