Solution to a Trigonometric integration

Question

I want to solve the following integration over a sphere of unit radius $$ \int_{0}^{2\pi} \int_{0}^{\pi} \sqrt{T_{1}^{2}\sin^{2}\theta\cos^{2}\phi+T_{2}^{2}\sin^{2}\theta\sin^{2}\phi+T_{3}^{2}\cos^{2}\theta}\sin\theta \,\mathrm{d}\theta \mathrm{d}\phi $$ Where $\theta$ runs from $0$ to $\pi$, $\phi$ runs from $0$ to $2\pi$ and $T_{1}$, $T_{2}$, $T_{3}$ are numbers between $0$ and $1$. I do understand that the $\phi$ integration is elliptic integral. However, is it possible to solve the $\theta$ integration and obtain the the elliptic integral form of $\phi$ integration?