Does the following iterated integral have a simple closed-form expression in terms of $z$?
$$ I = \int_0^\infty \int_0^\infty \sqrt{\frac{1 + x^2 y^2 + x^2 z^2 + y^2 z^2}{(x^2 + y^2 + z^2 + x^2 y^2 z^2)^3}} \, dy \, dx $$
where $x \in \mathbb{R^+}$ and $y \in \mathbb{R^+}$. Assume $z \in \mathbb{R^+}$ is a constant.
I tried substituting:
$$ u = \frac{1}{2} \left[ 1 + \frac{y^2(1 + x^2 z^2)-(x^2+z^2)}{y^2(1 + x^2 z^2)+(x^2+z^2)} \right] $$
which simplifies the integral to this:
\begin{align} I &= \int_0^\infty \frac{\pi}{2} \frac{1}{x^2+z^2} \operatorname{_2F_1} \left( -\tfrac{1}{2}, \tfrac{1}{2}; 1; 1{-}\left[ \frac{x^2+z^2}{1+x^2 z^2} \right]^2 \right) \, dx \\ &= \int_0^\infty \frac{1}{x^2+z^2} \operatorname{E} \left( \sqrt{1{-}\left[ \frac{x^2+z^2}{1+x^2 z^2} \right]^2} \right) \, dx \end{align}
$\operatorname{E}(k)$ here is the complete elliptic integral of the second kind (using elliptic modulus $k$, not the parameter $m=k^2$).
I can't seem to simplify this any further. Is there a way to proceed with this integral over $\operatorname{E}(k)$? (I have looked through tables of integrals of elliptic integrals, but nothing seems to match.) Or is there another way to simplify the iterated integral, e.g. via a substitution for both $x$ and $y$?
For the record, this integral arises when deriving a certain kind of prior distribution for the parameters of a particular probabilistic model.
Berger & Bernardo, 1992, "Ordered Group Reference Priors with Application to the Multinomial Problem" describe a sort of generalized Jeffreys prior which assumes a specific parameter ordering and grouping. Essentially, the "conditional Jeffreys prior" is derived for each group based on a conditional Fisher information matrix (with the parameters earlier in the ordering held constant), which is based on a marginal likelihood function (with parameters later in the ordering integrated out).
The integral above appears when deriving such a prior for a bivariate Bernoulli model with a specific parameterization and ordered grouping of parameters. For two binary variables $X$ and $Y$ with joint probability denoted e.g. $p_{11}$, consider the saturated model with three "odds ratio" parameters:
$$O_X=\left(\frac{p_{10} p_{11}}{p_{00} p_{01}}\right)^{1/4}, O_Y=\left(\frac{p_{01} p_{11}}{p_{00} p_{10}}\right)^{1/4}, O_{XY}=\left(\frac{p_{00} p_{11}}{p_{10} p_{01}}\right)^{1/4}$$
(These are similar to the classic log-linear parameters defined e.g. in Bishop, Fienberg, & Holland, 1975, Discrete Multivariate Analysis, but without the logs. I assume the integral here would also appear for the case with log-linear parameters.)
Assuming the ordered grouping: $(\{O_{XY}\},\{O_X,O_Y\})$, the integral above appears as part of the normalizing constant for the conditional prior $p(O_X,O_Y|O_{XY})$. Finding the joint prior $p(O_X,O_Y,O_{XY})$ then involves deriving $p(O_{XY})$ from a marginal likelihood.