Elliptic functions as inverses of Elliptic integrals

The claim as stated is not true. (E.g., if $R$ has only even powers of the second variable, the resulting function $f$ is the integral of a rational function.) What is true is that every general elliptic integral of this form can be expressed as a linear combination of integrals of rational functions and the three Legendre canonical forms (elliptic integrals of the first, second, and third kind). This is a classical result, and there are several different algorithms to reduce a general elliptic integral to this form, some of them implemented in common computer algebra systems.

A modern (freely available) reference with a list of classical references is here: B.C. Carlson, Toward Symbolic Integration of Elliptic Integrals, Journal of Symbolic Computation, 28 (6), 1999, 739–753