A decision procedure for finding closed forms for indefinite summation of polynomials, rational functions, quasipolynomials and quasirational functions is presented. It is also extended to deal with some non-hypergeometric sums with rational inputs, which are not summable by means of Gosper's algorithm. Discussion of its implementation, analysis of degree bounds and some illustrative examples are included. Copyright © 1993 Academic Press. All rights reserved.