$\mathbb{E} \left[ \frac{X}{c + X} \right]$ for known constant $c > 0$? In general, describing expectations of ratios of random variables can be tricky. I have a ratio of random variables, but thankfully, it's nicely behaved due to known structure. Specifically, I have a univariate random variable $X$ whose support is non-negative reals $\mathbb{R}_{\geq 0}$, and I want to compute
$$\mathbb{E} \Bigg[ \frac{X}{c + X} \Bigg]$$
where $c > 0$ is a known constant. Is there an exact expression for this expectation, perhaps in terms of the moments (or centered moments) of $X$?
Edit: The first-order Taylor series approximation is $\frac{\mathbb{E}_X[X]}{c + \mathbb{E}_X[X]}$ but I was hoping for something better / exact if possible.

Breaking down
$$
\frac{X}{c+X} = 1 - \frac{1}{1+X/c},
$$
and expanding in Taylor series, its expectation is
$$
E \left[ \frac{X}{c+X} \right] = 1 - E \left[ 1 - \frac{X}{c} + \frac{X^2}{c^2} - \frac{X^3}{c^3} + \cdots \right] \\
= E \left[ \frac{X}{c} - \frac{X^2}{c^2} + \frac{X^3}{c^3} - \cdots \right] \\
= \sum_{k=1}^\infty \frac{(-1)^{k-1}}{c^k} E[X^k]
$$
Now, that Taylor series only converges for $-c < X < c$, and so if the support of $X$ lies outside of this range, this expansion is invalid.
