Solutions of elliptic integrals and generalizations by means of Bessel functions

The elliptic integrals and its generalizations are applied to solve problems in various areas of science. This study aims to demonstrate a new method for the calculation of integrals through Bessel functions. We present solutions for classes of elliptic integrals and generalizations, the latter, refers to the hyperelliptic integrals and the integral called Epstein–Hubbell. The solutions obtained are expressed in terms of power series and/or trigonometric series; under a particular perspective, the final form of a class of hyperelliptic integrals is presented in terms of Lauricella functions. The proposed method allowed to obtain solutions in ways not yet found in the literature.