A method is proposed by which elliptic integrals can be integrated symbolically without information regarding limits of integration and branch points of the integrand thatis required in integral tables using Legendre's integrals. However, it is assumed that when all polynomials in the integrand have been factored symbolically into linear factors, the exponents of all distinct linear factors are known. The recurrence relations are one-parameter relations, all formulas are given explicitly, and the integral is eventually expressed in terms of canonical R-functions, with no increase in their number if neither limit of integration is a branch point of the integrand. It is the use of R-functionsrather than Legendre's integrals that makes it possible to carry out the whole processsymbolically. If (possibly complex) numerical values of the symbols are known, there are published algorithms for numerical computation of the R-functions.