Free computer algebra systems
SAGE
maxima
Axiom:
MathAction,
Axiom Portal,
Axiom developer,
OpenAxiom,
FriCAS
yacas
giac/xcas
GiNaC,
nestedsums,
gTybalt
sympy
mathomatic
jacal
Less known
Mathemagix
DoCon
eigenmath
Symbolic C++
HartMath
meditor
Specialized
NTL
pari/gp
GAP
Macaulay2
Singular
Cadabra
Not quite free
Aldor
Fermat
Form
CoCoA
GUI for many CASs
TeXmacs