How to solve the integral $$\int_{B(0,1)} xy(x+z)(y+z)dxdydz$$ where $B(0,1)=\{ (x,y,z) | x^2 + y^2 + z^2 < 1\}$?
I tried using spherical coordinates, but it turned out ugly. I then tried solving without any substitution but it wasn't so pleasant as well.
I think to use the substitution $u=xy, v = x+z, w = y+z$ but I can't determine the range. I also am not sure if it is a diffeomorphism.
Any suggestions?
Help would be appreciated
Hint. Use symmetry! After expanding the product, you will see that most of the terms to integrate are "odds" and their integral is zero over any ball centered at the origin. It remains to evaluate $$\int_{B(0,1)} x^2y^2\,dxdydz$$ Can you take it from here?