Deep Learning (DL) libraries (e.g., Pytorch) are popular in the development of AI applications. These libraries are complex and contain bugs. Researchers have proposed various bug-finding techniques for such libraries. Yet, there is much room for improvement. A key challenge in testing DL libraries is the lack of API specifications. Prior testing approaches often inaccurately model the input specifications of DL APIs, resulting in missing valid inputs that could reveal bugs or false alarms from invalid inputs. To address this challenge, we develop Centaur - the first neurosymbolic technique to test DL library APIs using dynamically learned input constraints. Centaur leverages the key idea that formal API constraints can be learned from a small number of seed inputs, and that the learned constraints can be solved using SMT solvers to generate valid and diverse test inputs for the API. We develop a novel grammar that represents first-order logic formulae over API parameters and expresses tensor-related properties (e.g., shape, data type, etc.) as well as relational properties between parameters. We use the grammar to guide a Large Language Model (LLM) to enumerate syntactically correct candidate rules, which are then validated using valid inputs. Further, we develop a custom refinement strategy to prune the set of learned rules to eliminate spurious or redundant rules. The learned constraints are then used to systematically generate valid and diverse inputs for the API by SMT solving (such as Z3) and a specialized sampling technique. We evaluate Centaur for testing PyTorch and TensorFlow. Our results show that Centaur generates constraints more accurately compared to prior approaches, namely DocTer and ACETest. In terms of coverage, Centaur covers 203, 149, and 9608 more branches than TitanFuzz, ACETest and Pathfinder, respectively. Using Centaur, we also detect 23 new bugs in PyTorch and TensorFlow, out of which 11 are already confirmed.