I ran into the following definite integral in 3 dimensions:
$$ \int_{0}^{\infty}dx_1 \int_{0}^{\infty}dx_2 \int_{0}^{\infty}dx_3 \ x_1^{\alpha_1} x_2^{\alpha_2} x_3^{\alpha_3} e^{-(x_1+x_2+x_3)}\left(x_1 x_2 A_3 + x_1 x_3 A_2 + x_2 x_3 A_1 \right)^{w} \ , $$
where $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{N}$ are integers bigger or equal than $0$, $A_1, A_2, A_3 \in \mathbb{R}^{\geq 0}$ are real numbers (positive or equal to $0$) and $w \in \mathbb{C}$ is a a complex variable.
The integral seems to be related to the $\Gamma$ function, or its generalizations: in fact, I tried to first switch to the set of coordinates given by
$$ x_i=y_i^2 \ , \ i=1,2,3 \ , \ y_{i} \in [0, \infty) $$
and then to spherical coordinates in 3-dimensions
$$ y_1=r \sin (\theta) \sin(\phi) \ , \ y_2=r \cos (\theta) \sin(\phi) \ , \ y_3=r \cos(\phi) \ , \ r\in [0, \infty) \ , \ \theta \in [0, \pi/2] \ , \ \phi \in [0, \pi/2]. $$
The integral in the $r$ variable can then be easily performed and returns a $\Gamma$ function. However, this leaves me with a complicated trigonometric integral, from which I have not been able to deduce a closed form formula.
My question is: is it possible to find a closed form formula for the integral above?