symbolic computation program/software

I recommend sageMath, also Mathematica and Maple

An easy to use is Maxima

You can to work with sage here: http://www.sagenb.org/, it is not necessary to install !!!

Both Mathematica and Maple are excellent.

It seems they have added symbolic computation toolbox for scilab (which is free). Since you are linux user, you might want to explore it. I have never personally used it though (I have used scilab itself and it does a good job of being a lilliputian Matlab)

http://packages.ubuntu.com/lucid/scilab-scimax-doc

Another very popular free program for symbolic computations is PARI/GP. It is aimed somehow at computations in number theory, but its functionality is not only restricted to this. You can read a little bit about it also in the wikipedia article.