In a paper I am reading:
Let $X$ and $Y$ be independent positive and non-degenerate random variables. Also let $S=X+Y$ and $R={X\over S}.$ If $R$ and $S$ are independent then: $$f_{R,S}({x\over{x+y}},x+y)=f_R({x\over{x+y}})f_S(x+y)=(x+y)f_X(x)f_Y(y).$$ Where $f$ is the density function.
How did we get the last equality?
The second equality doesn't actually rely on the independence of $R$ and $S$. Consider the following functions:
$$\begin{pmatrix} r\\ s \end{pmatrix} = \begin{pmatrix} \frac{x}{x+y}\\ x+y \end{pmatrix}:= G\left(\begin{pmatrix} x\\ y \end{pmatrix}\right).$$
Notice that this function is invertible, and its inverse is differentiable:
$$H\left(\begin{pmatrix} r\\ s \end{pmatrix}\right):= G^{-1}\left(\begin{pmatrix} r\\ s \end{pmatrix}\right) = \begin{pmatrix} rs\\ s-rs \end{pmatrix}.$$
So now we just apply a change of variable to $r = \frac{x}{x+y}$ and $s = x+y$:
\begin{align*} f_{R,S}\left(\frac{x}{x+y},x+y\right)&=f_{R,S}(r,s)\\ &= f_{X,Y}(x,y)\left|\det\frac{d(x,y)}{d(r,s)}\right|\\ &= f_X(x)f_Y(y)\begin{vmatrix}s&r\\-s&1-r\end{vmatrix}\\ &= f_X(x)f_Y(y)\left(s - sr + sr\right)\\ &=(x+y)f_X(x)f_Y(y). \end{align*}