From Courant John II, 4.6 Exercise 11:
Question statement:
$$\iiint (x+y+z)x^2y^2z^2 \,dx\,dy\,dz$$ on $$R: x+y+z\le 1, x\ge 0, y\ge 0, z\ge 0$$
I tried the change of variables $u=yz, v=xz, w=xy$ and got $$\frac12\iiint uvw\left(\dfrac1u+\dfrac1v+\dfrac1w\right)du \,dv\, dw$$
However I have no idea how to determine the region $R'$ over which I shall be integrating the function with respect to u,v, and w.
Since $x+y+z\le 1$, in particular let $x=0$, then by the A.M.-G.M. inequality, $u=yz\le (\dfrac{y+z}{2})^2\le \dfrac14$. Similarly, $v\le \dfrac14, w\le \dfrac14$. Additionally, substituting x,y,z by u,v,w, I got $\sqrt{uvw}\left(\dfrac1u+\dfrac1v+\dfrac1w\right)\le 1$. By A.M.-G.M., $\sqrt[6]{uvw}\le \dfrac1{3^6}=\dfrac1{729}$.
How do I proceed?
Suggest you to use the change of variables $(x,y,z)\mapsto (u,v,w)$ such that
$$u=x+y+z\,,\,uv=x+y\,,\,uvw=x$$
That is, $$x=uvw\,,\,y=uv(1-w)\,,\,z=u(1-v)$$
Verify that $$x+y+z\le 1\,,\,x,y,z\ge 0\implies 0<u,v,w<1$$
And absolute value of jacobian determinant is $$|J|=u^2v$$
Hence,
$$\iiint_R (x+y+z)x^2y^2z^2\,dx\,dy\,dz=\int_0^1 u^9\,du\int_0^1 w^2(1-w)^2\,dw\int_0^1 v^5(1-v)^2\,dv$$