The Risch "algorithm", in this sense, is an implementation of the Risch decision procedure; in particular, the Risch-Bronstein-Trager algorithm. Axiom's version was written, in large part, by Bronstein, himself. Axiom's implementation is complete in the sense of capturing the entire Risch decision procedure for elementary functions: it either returns the antiderivative or returns a proof that no such antiderivative exists.

However, while elementary functions are extremely important, they aren't the whole story for functions; in that sense, Axiom is far weaker than, say, Mathematica.