Unexpected integration result

Questions
,
1

Consider the following integral:

Integrate[
 1/Sqrt[(x - u)^2 + v^2 + w^2], {u, v, w} \[Element] 
  Ellipsoid[{0, 0, 0}, {a, a, b}], 
 Assumptions -> 
  a \[Element] Reals && a > 0 && x \[Element] Reals && x > a && 
   b \[Element] Reals && b > 0 && a > b]

It computes the gravitational potential generated by a body of ellipsoid shape. Mathematica 13.1 gives the result 0.
I don’t know what the correct result is, but not 0 for sure!

2

Yes, that is strange, even with numerical parameters:

With[{a = 2, b = 1, x = 3},
 Integrate[1/Sqrt[(x - u)^2 + v^2 + w^2],
  {u, v, w} \[Element] Ellipsoid[{0, 0, 0}, {a, a, b}]]]

With a sphere the result is sensible, though:

With[{a = 1, b = 1, x = 3},
 Integrate[1/Sqrt[(x - u)^2 + v^2 + w^2],
  {u, v, w} \[Element] Ellipsoid[{0, 0, 0}, {a, a, b}]]]