Deep Learning (DL) libraries (e.g., PyTorch) are popular in AI development. 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 missed valid inputs that could reveal bugs or false alarms due to 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 automatically generated seed inputs, and that the learned constraints can be solved using SMT solvers to generate valid and diverse test inputs. We develop a novel grammar that represents first-order logic formulae over API parameters and expresses tensor-related properties (e.g., shape, data types) as well as relational properties between parameters. We use the grammar to guide a Large Language Model (LLM) to enumerate syntactically correct candidate rules, validated using seed inputs. Further, we develop a custom refinement strategy to prune the set of learned rules to eliminate spurious or redundant rules. We use the learned constraints to systematically generate valid and diverse inputs by integrating SMT-based solving with randomized sampling. We evaluate Centaur for testing PyTorch and TensorFlow. Our results show that Centaur's constraints have a recall of 94.0% and a precision of 94.0% on average. In terms of coverage, Centaur covers 203, 150, and 9,608 more branches than TitanFuzz, ACETest and Pathfinder, respectively. Using Centaur, we also detect 26 new bugs in PyTorch and TensorFlow, 18 of which are confirmed.