Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by Kotz and Nadarajah)
\begin{equation}
 f(x)=\frac{\Gamma((\nu+p)/2)}{(\pi)^\frac{p}{2}\Gamma(\nu/2)|R|^\frac{1}{2}}
\left[ 1+\frac{1}{\lambda}(x-\mu)^T R^{-1} (x-\mu) \right]^{-(\nu+p)/2},
\end{equation}
where $x \in R^p$.


I want to find the expectation, under $f(x)$ of something similar to the density, i.e.,
I want to evaluate


\begin{equation}
 \int\_x f(x) \left[ 1+\frac{1}{\lambda\_2}(x-\mu\_2)^T R\_2^{-1} (x-\mu\_2) \right]^{-\nu\_2} dx,
\end{equation}
where $R\_2$ is a covariance matrix, $\mu\_2$ is a vector, and $\lambda\_2>0$.


Is there a closed form expression for this integral for $x \in R^p$? In 1-d?



Expectation under a t-distribution
I am not sure a closed-form expression is known in the literature, but I found this [paper](http://www.springerlink.com/index/6833356842Q37836.pdf) that describes other properties of that integral without solving it.


