The integral in the question appears in the solution to the orbit of a particle subjected to a central force. It is written in Goldstein's Classical mechanics that the solution is possible in terms of circular trigonometric functions without providing any reason. The wiki page https://en.wikipedia.org/wiki/Exact_solutions_of_classical_central-force_problems refers to Whittaker which says that the expression under the square root must be quadratic at most. I don't understand why it should be quadratic at most, i.e., why other values of n would make the integral stop having solutions with elementary functions only, what's the guarantee that no other integral value of n is possible?
Any help/reference will be appreciated. Thanks in advance.
As for as I know, the only values of $n$ for which we have closed form expressions are $$\left\{\color{red}{-2},-1,\color{red}{0},\frac{1}{2},\color{red}{1},\frac{3}{2},\color{red}{2},3,4\right\}$$
Only the "red" ones leads to elementary functions; the other ones lead to very complex elliptic integrals of the first, second and third kinds (complete and incomplete).
For the other values, have a look at the link provided by @Jean-Marie.