The telescoping-languages approach achieves high performance from applications encoded as high-level scripts. The core idea is to pre-compile underlying libraries to generate multiple variants optimized for use in different possible contexts including different argument types. We have previously developed a type inference algorithm that enables type-based specialization. The algorithm infers types over untyped library procedures before actual inputs are known. Type inference is necessary both to determine the minimum number of variants needed to handle all possible uses of the library procedure as well as to statically determine, for each variant, which optimized implementations should be dispatched at each call location. In this paper, we extend the type inference algorithm to handle user-defined types in an object-oriented scripting language such as Python. Variants will be generated in order to eliminate dynamic dispatch where possible. This will allow for faster runtimes as well as enable more cross-method optimizations. Two problems that arise when handling user-defined types are the lack of a bound on the number of types and the possibility of extensions to the type hierarchy after type inference. We address these problems and describe the type inference solution.