Finding distribution of sum of two dependent variables

I am trying to find the distribution of the sum of two random variables $$(x_1,x_2)$$ which have the following distribution:

$$f(x_1,x_2) = \begin{cases}2 ( x_1 + x_2) & 0 \le x_1 \le x_2 \le 1 \\ 0 & \text{otherwise} \end{cases}$$

The expression I thought would work was

TransformedDistribution[
x1 + x2, {x1, x2} \[Distributed]
ProbabilityDistribution[2 (x1 + x2), {x1, 0, x2}, {x2, x1, 1}]]


But this only returns the expression unevaluated. What would be the proper method to calculate the result?

I figured it out -- the expression is representing the distribution object. To get the formula, I had to call PDF on the distribution. The code would then look like
PDF[TransformedDistribution[

$$\begin{cases} -((-2+w)w) & 1 < w < 2 \\ w^2 & 0