Some integrals with problematic evaluation

[parisse]

Step by step was not working when the last firmware was released, and it is still in early testing stages today, you can't expect to have it inside the Prime today.
I just checked with the web version of Xcas with Firefox, int(x^1/(a^5+x^5)), int(x^2/(a^5+x^5)), int(x^3/(a^5+x^5)) return the antiderivative without any rootof. Make sure to clear your browser cache.
Simplify does not mean you will get a simpler answer, because that can not be well defined, sometimes you need a factored expression, sometimes an expanded one, sometimes with sin, sometimes with cos, etc. That's why Giac provides a lot of rewriting commands. The simplify command should perhaps be renamed, it is a more powerful version of normal in the sense that it handles relation between transcendental functions like sin^2+cos^2=1. normal handles rational expressions and algebraic extensions, ratnormal handles only rational expressions. regroup is completly different, it does not rewrite the expression as an irreducible fraction, it just collects some terms without modifying much the structure of the expression.
You will always find some expressions that are better simplified or that return a simpler integral on one CAS than on an other one. I don't know for sure what algorithms the nspire is using because their CAS is not opensource, but I think they do more table-lookups than I do (they have probably lookup for common US textbooks exercices) and did not implement more advanced algorithms like Hermite reduction for rational fractions or the Risch algorithm : for example int(1/(x^4+1)^10) is slow or int((2x^2+1)*exp(x^2)) is left as is.