Linear combination of coordinates of random unit vector

Let $$v\in \mathbb{R}^n$$ be uniformly distributed on the unit sphere. Let $$\lambda_1,...,\lambda_n$$ be given real numbers. What is the distribution of $$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$ Does it happen to belong to any known family of distributions? I think this is a very flexible way to model the distribution with compact support. When $$n=2$$, $$X$$ is just the celebrated arcsine distribution supported on $$(\lambda_{\min},\lambda_{\max})$$. What about for general $$n$$? I also think $$X$$ can capture the ''spreadness'' of the sequence $$\lambda_1,..,\lambda_n$$.

The distribution of $$X$$ is the distribution of the ratio $$\frac{\sum_{i=1}^n\lambda_iZ_i^2}{\sum_{i=1}^nZ_i^2}$$ of two quadratic forms in iid standard normal random variables $$Z_1,\dots,Z_n$$ (because the distribution of $$(v_1,\dots,v_n)$$ is the same as that of $$(Z_1,\dots,Z_n)\big/\sqrt{\sum_{i=1}^nZ_i^2}$$). The distribution of such ratios was studied by Gurland; also see e.g. Watson.