The integral is \begin{equation} I = \int\limits_{0}^{\infty}\frac{\rm{d}x}{\sqrt{x}(1 + 3x^2)} \end{equation} I cannot figure out how to solve this applying the Residue theorem, as I have not a closed curve... My function is neither odd or even, any ideas?
EDIT:
As I'm still looking for a complete solution I ask if someone can show how to compute the residue for the singularities on the positive half plane ($\Im{z}>0$), because anyway I try to calculate them I get the wrong result. Thanks in advance.
If you perform the substitution $x=t^2$ you are left with $$ \int_{0}^{+\infty}\frac{2\,dt}{1+3t^4} = \int_{\mathbb{R}}\frac{dt}{1+3t^4} $$ where the last integral equals $2\pi i$ times the sum of the residues of $\frac{1}{1+3t^4}$ at the simple poles in the upper half plane. That is a straightforward computation.