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To address your reliability or power question briefly from a different vantage point, I think it depends a lot on what you are doing. If you need the most symbolic antiderivatives you can, Maple or Mma are probably better; on the other hand, many people in "real work situations" use Maxima as a standalone program very, very effectively for symbolic DEs and such. If you are doing more numerical work, my understanding is that the Numpy/Scipy/Matplotlib stack, or the GSL (both in Sage), are pretty darn effective and not necessarily "buggy" at all. If you are doing graph theory or serious number theory, you shouldn't even be asking the question of which package to use. But either of them (or Maple) will probably do 99% of what you need to do, correctly, in most cases.

Matlab is a different situation, because of the enormous amount of third-party code and drivers available (only) for it. Octave might be a free option, but it would really depend on what you need to do.

In the end, you'll have to figure things out for yourself. If your department head is too cheap (to be crass, which is unavoidable in your situation) to buy you a Mathematica license, then they are really putting their money where their mouth is, and you should use Sage.

There are plenty of bugs in any mathematical software, and a quick internet search will find them for you. You should never ever implicitly trust output from a computer in any case, without some other means of checking it (such as "plugging in the answer"). Even when Euler proved that $\zeta(2)=\frac{\pi^2}{6}$ or that $\gamma$ exists, he also checked it by hand to an insane number of digits. Thankfully, we don't have to do that to check our work.