distribution of two variables each follows scaled chi-square

Question

Let $S$ is a test statistics and $w_1$ is a weight. $w_1 S$ follow a scaled $ c_1 \chi^2_{d1}$. Then let $w_2 S$ follow a scaled $ c_2 \chi^2_{d2}$.

What is the distribution of $w_1 S +w_2 S$ ?

We can re-write this as; $T_1=w_1 S $ and $T_2=w_2 S $, then what is the distribution of $ \pi T_1 + (1- \pi)T_2 $ ? where $\pi$ is a constant.

Answer 1

This is going to result in a Generalized $\chi^2$ distribution. Not something particularly nice.