Let me try
Rewirte the integrand :
Then substitute
The integral becomes :
Sub.
The integral
But we have
You have totally annihilated it!
Now that I've the answer, I see that it could have been done it in the following way:
Rewrite the integrand:
[tex] \displaystyle = \frac{a(a^2-x^2)}{\sqrt{(a^2+ax+x^2)}\sqrt{(a^2-ax+x^2)}(a^2-ax+x^2)} [/tex]
Let (One would have to employ clairvoyance to spot that substitution without knowing the answer beforehand )
Then
The integrand cancels to 1 (since the two products are inverse to each other), therefore .
Hence