Symmetric standard elliptic integrals are of the form
where , parameters , , are nonnegative and distinct, and may eventually coincide with . The author constructs the complete asymptotic convergent expansions of these integrals when either or are large (tend to ). The distributional approach is used for a new derivation of the asymptotic expansion of the generalized Stieltjes transforms, which is applied to the integrals above. Coefficients of these expansions are computed by recurrence. Moreover, accurate error bounds for the remainder are supplied. Numerical results show that the rate of convergence of the asymptotic series increases with the large parameter.