[Axiom-math] Axiom's integration non-deterministic?

[Igor Khavkine]

Start up Axiom, and type in the following integral

  integrate(sqrt(1+x^(-2/3)),x)

Then type in the same integral again. Maybe try the same integral a
few more times. Do you get the same answer each time?

I don't. I see the same behavior from axiom-20050901-4.1 (Debian sid)
and axiom-20050201-1 (Debian sarge).

I get two possible answers, one is (x^(3/2)+1)*sqrt(x^(3/2)+1) and the
other one is a much more complicated radical expression. Ironically,
the longer expression takes less time to compute. Differentiating both
answers gets me back to the original integrand. Also, their curves
coincide when plotted. It seems that each time the answer is correct.
However, the shorter one is obviously more attractive because it looks
simpler.

Clearly, Axiom takes two different paths through the integration
algorithm, even when give identical input. What is the cause of the
branch? Is there a non-deterministic step somewhere in the algorithm?

Incidentally, is there a canonical form for radical expressions in
which the two forms of the answer can be compared and directly shown
to be the same?

Thanks in advance.

Igor