[math-fun] computational complexity of "reciprocated subset sum"

Given N numbers x1, x2, ..., xN, I want to compute the sum 1/(x1+x2+..+xN) + 1/() + 1/() ... where my sum has 2^N - 1 summands, one for each nonempty subset of {1,2,3,...,N}, and the denominators are the sums of the x's with indices in each subset. Obviously, my sum can be computed in O(2^N) operations. I want to know if there is any way to do it faster.