I'm vaguely in the "market" for a new computer algebra system (Mathematica, Maple, Sage, SymPy, etc etc) and I am curious what experience you all might have with these types of programs. In the past I've used Mathematica, and once you get over the odd aspects of the Language (and hefty license fee), it is a pretty great piece of software. Nonetheless, now that my old student license has expired and I would have to buy a new one (though through my school I can get it at a steep discount) I'm looking at some alternatives. Sage seems particularly appealing since it is open source and apparently well documented, and of course SymPy is powerful although that as stand-alone as these other choices.

There are tons of other options as well, with a wide range of functionality (it really is amazing how much these things can do). My main use cases with these types of things are either: a) algebraic manipulation, b) linear algebra, c) visualization. For these three it does really seem that Mathematica is at the forefront, but I'm all ears for other opinions.

I'm curious, if you have used any or all of these in the past, what were you using them for and what is your opinion?