Accelerating indefinite hypergeometric summation algorithms

Let K be a field of characteristic zero, x an independent variable, E the shift operator with respect to x, i.e., Ef(x) = f(x + 1) for an arbitrary f(x). Recall that a nonzero expression F(x) is called a hypergeometric term over K if there exists a rational function r(x) ∈ K(x) such that F(x + 1)/F(x) = r(x). Usually r(x) is called the rational certificate of F(x). The problem of indefinite hypergeometric summation (anti-differencing) is: given a hypergeometric term F(x), find a hypergeometric term G(x) which satisfies the first order linear difference equation (E − 1)G(x) = F(x). (1) If found, write Σ_x F(x) = G(x) + c, where c is an arbitrary constant.