“Axiom is a general purpose Computer Algebra system. It is useful for doing mathematics by computer and for research and development of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler.”

There is also an interesting Rosetta Stone which offers translations of many basic operations for several computer algebra systems, including Maxima.

“GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory.”

“Jasymca is a symbolic calculator written for mobile phones and PDAs. It solves and manipulates equations, handles basic calculus problems, and provides a few more typical functions of computer algebra systems. The syntax is loosely related to GNU-Maxima.”

“REDUCE is an interactive system for general algebraic computations of interest to mathematicians, scientists and engineers.”

“SINGULAR is a Computer Algebra System for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry, and singularity theory.”

“YACAS is an easy to use, general purpose Computer Algebra System, a program for symbolic manipulation of mathematical expressions. It uses its own programming language designed for symbolic as well as arbitrary-precision numerical computations.”

- Oberwolfach References on Mathematical Software: http://orms.mfo.de/
- Free mathematical and computational software directory: http://cadadr.org/fm/

“ARIBAS is an interactive interpreter for big integer arithmetic and multi-precision floating point arithmetic with a Pascal/Modula like syntax. It has several builtin functions for algorithmic number theory like gcd, Jacobi symbol, Rabin probabilistic prime test, factorization algorithms (Pollard rho, elliptic curve, continued fraction, quadratic sieve), etc.”

→ http://www.mathematik.uni-muenchen.de/~forster/sw/aribas.html

“The fundamental package needed for scientific computing with Python is called NumPy. This package contains a powerful N-dimensional array object, sophisticated (broadcasting) functions, tools for integrating C/C++ and Fortran code, and useful linear algebra, Fourier transform, and random number capabilities.”

“GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with Matlab.”

“PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions.”

“R is a free software environment for statistical computing and graphics. It compiles and runs on a wide variety of UNIX platforms, Windows and MacOS.”

Lots of links there.

→ http://en.wikipedia.org/wiki/List_of_computer_algebra_systems

A very good starting point to learn about symbolic computation and computer algebra systems.