Using spherical coordinates for triple integral $ \int_{0}^3 \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}}  \frac{\sqrt{x^2+y^2+z^2}}{1+x^2+y^2+z^2} \ dz \ dy \ dx$
Using spherical co-ord's this becomes : 
$ \int_{0}^{2\pi} \int_{0}^\pi \int_{0}^3 \frac{r^3\sin(\theta)}{1+r^2} \ dr \ d\theta \ d\phi $
Is this correct? If it is how do I carry on from here? 

Your limits aren't quite correct. The spherical region you are integrating over is only in the first octant. Therefore $\theta$ should range from 0 to $\pi/2$ and $\phi$ should range from $0$ to $\pi/2$. 
