= Solution
Assume the spheroid is homogeneous, as implicit in the stated moments. Put $s=\sqrt{1-e^2}$ and map a uniform solid sphere by $(X_1,X_2,X_3)\mapsto(X_1,X_2,sX_3)$, preserving total mass by adjusting density. For the sphere, symmetry and its moment $2Ma^2/5$ give $\int X_i^2\,dm=Ma^2/5$ for each coordinate. The transformed squared-coordinate integrals are consequently $Ma^2/5,Ma^2/5,s^2Ma^2/5$. Reflection symmetry makes the off-diagonal inertia entries vanish. Thus
$$
\boxed{I_1=I_2=\frac{Ma^2}{5}(1+s^2)=\frac25Ma^2(1-e^2/2),
\qquad I_3=\frac25Ma^2.}
$$
The original PDF gives the correct solid-sphere factor $2/5$, rather than the converted TeX's $2/3$.
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