Components of an Electric field due to a dipole

[forensics409]

Homework Statement

The problem is: Show that the components of $$\vec{E}$$ due to a dipole are given at distant points, by E_x=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{3pxz}{(x^2+z^2)^{5/2}}$$ and E_z=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{p(2z^2-x^2)}{(x^2+z^2)^(\frac{5}{2})}}$$


http://physweb.bgu.ac.il/COURSES/PHYSICS2_B/2009A/homework/Homework-2_files/image006.jpg

Homework Equations

E=$$\frac{1}{4\pi\epsilon{o}}$$ $$\frac{Q}{r^2}$$
p=qd

The Attempt at a Solution

I have tried to break the fields of each one into vector components and add the components, however, it got really messy really quickly and after simplifying it a bit i got a ridiculous equation for just the x component, i had no clue where to go and gave up on even try to get the z component.

E_{x=$$\frac{q}{4\pi\epsilon{o}}$$ $$\frac{(x^2+(z+\frac{d}{2})^{2})^{\frac{3}{2}}-(x^2+(z-\frac{d}{2})^{2})^{\frac{3}{2}}}{((x^{2}+z^{2})^{2} + (\frac{d^{2}x^{2}}{2}-\frac{d^{2}z^{2}}{2}+\frac{d^4}{16}))^{\frac{3}{2}}}$$}

 

[AvroArrow]

I can't figure out how to simplify this expression to get the one given, any help would be greatly appreciated.