On the relation between the non-flat cosmological models and the elliptic integral of first kind

Recently we have found an analytic integration formula that describes the dependence of the luminosity distance on the redshift for the flat cosmological model with a non-zero cosmological constant. The purpose of this article is to search for a similar relation for the non-flat models. A Taylor series was used. The elliptic integral of the first kind can indeed be applied. The result shows that a combination of the numerical integration with the previous analytic formula can also be useful for the nearly flat cosmological models.