Maxima - Maxima is a fairly complete computer algebra system written in Lisp with an emphasis on symbolic computation.
Cadabra - Cadabra is the name of a universal computer algebra system designed for those users looking for the solution to problems encountered in the filed theory.
Sage Math - Sage is a free open-source mathematics software system licensed under the GPL.
MATLAB - A high-level language and interactive environment for numerical computation, visualization, and programming
Macaulay2 - Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra, whose creation has been funded by the National Science Foundation since 1992.
Magma - A software package designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics.