Wikipedia luckily provides some context, as the linked page is bit spartan: > In 2007, Axiom was forked twice, originating two different open-source projects: OpenAxiom[3] and FriCAS,[4] following "serious disagreement about project goals".[5] The Axiom project continued to be developed So FriCas is one of the three main branches of Axiom. But what is Axiom? > Axiom is a free, general-purpose computer algebra system.…
The current Axiom effort is directed toward proving the algorithms. The research combines computer algebra with proof theory. A survey done last year shows that there is almost no overlap between these two fields of computational mathematics.
Computer algebra IS mathematics. The present state of the art is "works-for-my-case" implementations. That is fine for the short term but Axiom focuses on the "30 Year Horizon". Early computer algebra systems did integration "by heuristics". Systems like Axiom use the Risch algorithm.
The next generation systems should use proven algorithms. At this time this is still an open research area. Axiom was originally a research platform (at IBM Research) and, as open source, still retains its research focus. Research takes time.
This is a difficult research problem. Not everyone agrees it can or should be done. "Works-for-my-case" implementation is easier.
Axiom and Fricas are not in competition. We simply have different goals. The Fricas people are doing excellent work. I would encourage you to try the software.
Tim Daly (Axiom Lead Developer)