Which tool can I use to (numerically?) evaluate
$$ \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty x^2\exp\left(-a\sqrt{x^2+y^2+z^2}\right) dx ~ dy ~ dz,~a>0 $$
without having to transform to spherical coordinates first?
I tried SymPy online without success:
a = symbols('a', positive=True)
x, y, z = symbols('x y z')
integrate(x**2*exp(-a*sqrt(x**2+y**2+z**2)), (x, -oo, oo), (y, -oo, oo), (z, -oo, oo))
If I transform to spherical coordinates, it evaluates without problems:
a = symbols('a', positive=True)
r, theta, phi = symbols('r theta phi')
integrate(r**2*sin(theta)*r**2*sin(theta)**2*cos(phi)**2*exp(-a*r), (r, 0, oo), (theta, 0, pi), (phi, 0, 2*pi))
$$ \frac{32\pi}{a^5} $$