AlbertRich
commented
4 years ago Sorry, Rubi can't integrate rational functions involving quartics in the denominator, unless they can be factored.
The currently available Rubi 4.16.1 easily integrates (x^4 + x^3)/(x^4 - x^3), (x^4 + x^3)/(x^4 - x^2) and (x^4 + x^2)/(x^4 - x^3).
Albert
1/(1 + 2 x + x^3 - x^4)
Integrate[1/(1 + 2 x + x^3 - x^4), x] // Normal // ToRadicals // Simplify
gives sensible extensive explicit answer in elementary functions.
The same is with 1/(1 + 2 x - x^4) 1/(x^4 + 2 x + 4)
Mathematica easily solves this: (x^4 + x^3)/(x^4 - x^3) with result x + 2 Log[-1 + x] but Rubi cannot. The same for (x^4 + x^3)/(x^4 - x^2) and (x^4 + x^2)/(x^4 - x^3)