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I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically:

$F(x) = \frac{1}{w} \sum_{i=1}^x\frac{N(N+(w-1)i)}{i(N-i)}w^{i-x}$

This can be viewed as a Riemann sum for this integral:

$G(x) = \frac{1}{w}\int_{t=1}^x\frac{N(N+(w-1)t)}{t(N-t)}w^{t-x}dt$

Throwing either the integral or the sum into Wolfram Alpha returned some complicated function of Ei(x), the exponential integral function, that I don't have any intuition about, and testing it now doesn't return anything. I haven't seen any rules for dealing with the product of a rational function with an exponential in integral. I'm feeling stuck, can anyone help me out?

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