Answer
$\dfrac{312,500 \pi}{7}$
Work Step by Step
Conversion of rectangular to spherical coordinates is as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and
$\rho=\sqrt {x^2+y^2+z^2}$;
$\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$\
Here, we have $(x^2+y^2+z^2)^2=\rho^4$
$I=\int_0^{\pi} \int_{0}^{2 \pi}\int_{0}^{5} (\rho^4) (\rho^2) \sin \phi d\rho d\theta d\phi$
or, $I=\int_0^{\pi} \int_{0}^{2 \pi}\int_{0}^{5} \rho^6 \sin \phi d\rho d\theta d\phi$
or, $I=\int_0^{\pi} \sin \phi d\phi \times \int_{0}^{2 \pi}d\theta \times \int_{0}^{5} \rho^6 d\rho $
or, $I=[-\cos \phi]_0^{\pi} [\theta]_{0}^{2 \pi} [\dfrac{\rho^7}{7}]_0^5$
or, $I=(2 \pi) (1+1)
\times \dfrac{78125}{7}$
or, $I=\dfrac{312,500 \pi}{7}$
