Ratio of Noncentral Chi Squared

Let $X$ and $Y$ be independent random variables of noncentral chi squared distributions (https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution)$\chi_a^2(\delta)$ and $\chi_b^2(\eta)$ respectively. If $\eta = 0$, then $Z=\frac{b X}{a Y}$ is known to have noncentral F-distribution $F_{a,b}(\delta)$. In particular, when $\delta = 0$ also, it will be the central F-distribution $F_{a,b}$.

Questions :

1. If $\eta > 0$ and $\delta = 0$, then $\frac{a Y}{b X} = \frac{1}{Z}$ has noncentral F-distribution $F_{b,a}(\eta)$. A transformation on this density will yield the density of $Z$ in this case. Does this density have a name?
2. If $\eta > 0$ and $\delta > 0$, what is the most convenient way to describe the distribution of $Z$ (in terms of other known distributions) ? If there is no convenient way, what would be a closed form expression of its density?