Suggestions on how to approach the following inetgral I'm trying to solve the following integral:
$$\int_{0}^{\infty} \frac{\left( x^2 - (x_0 + y)^2 \right)^{2}}{\delta x_0^2 x^2 + \left(x^2 - (x_0 + y)^2 \right)^2} e^{-y/y_0} \ {\rm{d}}y$$
All my attempts so far are fairly elementary manipulations and just using integration by parts. I'm not having much success.
Can anyone suggest an approach, or some integral identities/theorems that might help?
 A: By polynomial division:
$$\frac{(x^2-(x_0+y)^2)^2}{\delta x_0^2x^2+(x^2-(x_0+y)^2)^2} = 1 - \frac{\delta x_0^2x^2}{\delta x_0^2x^2+(x^2-(x_0+y)^2)^2}$$
And hence the original integral is transformed into:
$$\left(\int_0^\infty e^{-y/y_0}dy\right) - \left(\delta x_0^2x^2\int_0^\infty\frac{e^{-y/y_0}}{\delta x_0^2x^2+(x^2-(x_0+y)^2)^2}dy\right) $$
Focusing on the rightmost integral, let $ w=x_0+y $ and hence:
$$\int_{x_0}^\infty\frac{e^{-(w-x_0)/y_0}}{\delta x_0^2x^2+(x^2-w^2)^2}dw$$
Bring out the multiplicative constant from the exponential term:
$$e^{x_0/y_0}\int_{x_0}^\infty\frac{e^{-w/y_0}}{\delta x_0^2x^2+(x^2-w^2)^2}dw$$
Now there is 4th degree polynomial on the denominator which will have four complex roots. By partial fraction decomposition, this integral can be broken down into four terms of the form:
$$ I= \int_{x_0}^\infty \frac{e^{-w/y_0}}{x+a}dw$$
I presume that $\delta>0$ and hence $a$ is necessarily complex.
The indefinite integral of the preceding definite integral is:
$$ e^{a/y_0}Ei\left(-\frac{x+a}{b}\right)$$
Where Ei(x) is the exponential integral.
