3D Change of Variables

Our math professor gave us this take-home project:

Consider a solid in the shape of the region D inside the surface

x^2 / (z^3 - 1)^2 + y^2 / (z^3 + 1)^2 = 1

If the density of the solid at the point (x,y,z) is x^2 + y^2 + z^2 then determine the mass of this solid. A GOOD CHANGE OF VARIABLES WILL HELP.

I understand how to do the problem but I can't get a change of variables that works well. I've tried cylindrical and spherical and many other random ones. Can anyone suggest a good change of variables to use? My teacher said that the cross sections for integration are in the shape of ellipses. Thank you!
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 Recognitions: Homework Help Science Advisor In problems which are of the form u^2 + v^2 = 1, you will frequently find that a useful change of variable is something like u = r\cos(\theta) and v= r\sin(\theta). Did you try this? Carl
 Recognitions: Homework Help If you cross-multiply the given equation, you arrive at $$x^2(z^3+1)^2+y^2(z^3-1)^2=(z^6-1)^2$$ and so it would seem that a likely useful change of variables would be: $$u=x(z^3+1),v=y(z^3-1),w=z^6-1$$ so that the equation then becomes $$u^2+v^2=w^2$$ which is a cone in uvw-space; but I haven't figured out just what the solid in xyz-space is: what is that "region D" that you were given?

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