I’m supposed to calculate the following integral by the divergence theorem:

$$

iint_S frac{x^2 +y(x+z^2)+z(x+y^2)}{sqrt{16x^2 +y^2 +z^2}} dS

$$

where $S$ is the ellipsoid $4x^2 +y^2 +z^2 =1$.

So the first thing I did, was to find out the normal unit vector $vec{n}$ in order to find out what our function $vec{F}$ is. Lucky I realised that $vec{n}=(2x,y,z)$ so, clearly we have:

$$

vec{F}=frac{left(x/2,(x+z^2),(x+y^2)right)}{sqrt{16x^2 +y^2 + z^2}}

$$

After that, I can argue that the flux I’m calculating at that surface will be the same as if I used a second function:

$$

vec{F}_2=frac{left(x/2,(x+z^2),(x+y^2)right)}{sqrt{12x^2 +1}}

$$

Now I simply need to find the divergence of that, which is $frac{1}{2 (1 + 12 x^2)^{3/2}}$

**The problem is** the tripple integral of this function seems rather complicated over the volume of the ellipsoid, at least in cylindrical coordinates.

Are there simpler ways to solve this problem? Am I not realising a simpler way? I appreciate any tips! Or solutions!