Contour integral with pure imaginary pole of order two : $\int_{0}^{\infty}\frac{x^2}{(x^2+9)^2(x^2+4)}dx$

by dan   Last Updated July 12, 2019 06:20 AM

I am trying to calculate $\int_{0}^{\infty}\frac{x^2}{(x^2+9)^2(x^2+4)}dx$. I want to use the upper half-circle as the contour (since the integrand is even), yet the function as poles in $2i , 3i$, but the pole $3i$ is an order $2$ pole, I couldn't find a way to calculate the residue of $3i$ in order to calculate the integral. Is there a way to split the integrand to $\frac{A}{x^2+9}$ + $\frac{B}{x^2+9}$ + $\frac{C}{x^2+4}$? or simple way to calculate the residue to $x=3i$?

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