Is there a name for the class of functions which are infinitely integrable in elementary functions?

No, unfortunately there is not name for these functions. In general, the integral of an non-fractional elementary function will result in another elementary function. Polynomials are the name for $ax^2 + bx + c$ sort of things with this being a three term polynomial. In general, I think the term you want is a non-fractional elementary function. Really, this all depends on what you define as elementary. If you include natural logs, and exponents then you would use that as a descriptive term. In general, functions are classes according to their form and basic characteristics, not according to how they react within an integral or derivative.