The most powerful computer algebra systems available are normally commercial programs (i.e. Matlab, Mathematica or Maple). However, there is an old (and very powerful) GPL computer algebra system called Maxima, which I recently discovered. The newer versions come bundled with a WXWindows GUI called WxMaxima, which makes Maxima a friendlier environment. Maxima is capable of many of the standard numerical procedures, such as differentiation, integration, solving systems of linear equations, and matrix algrebra, and even has inline plot integration with Gnuplot and equation rendering with LateX! I would recommend giving Maxima a try.