Axiom is my primary Computer Algebra System (CAS = symbolic manipulator) for three reasons:
The Axiom programming language is elegant and powerful, and gives a great deal of control on what's being done and how.
Axiom runs fast.
Axiom has carefully implemented algorithms.
Granted, Axiom is less "feature rich" than competitors (Mathematica, Maple, Maxima). Given that the additional features found in other CAS are not always bug-free, I'm OK with a smaller feature set.
Not everyone's cup of tea for sure but deserving of a place in the arsenal of people who can code.
P.S. Axiom has a Sage (www.ohloh.net/p/sage) interface:
(Disclaimer: On the basis of minor contributions made many many years ago, I am listed as a contributor.)