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I have a problem solving the following integral : $$\int_{0}^{\infty }\frac{e^{-\upsilon\;x}\:x^{\mu -1}}{\left ( x+a \right )^{m}\left ( x+b \right )^{n}}\:dx$$ This integral can by solved by using partial fraction decomposition for the following: $$\frac{1}{\left ( x+a \right )^{m}\left ( x+b \right )^{n}}$$ where $\mu$, $m$, and $n$ are integers numbers, $a> 0$ and $b> 0$.

However, I couldn't find a general formula for the partial fraction decomposition above to simplify the integral.

Looking for a solution for the integral using partial fraction decomposition or any other methods. Thanks in advance.

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It is just a guess, but maybe you should approach this problem in the following way \begin{align*} \frac{1}{(x+a)^{n}(x+b)^{m}} & = \frac{1}{b-a}\frac{(x+b) - (x+a)}{(x+a)^{n}(x+b)^{m}}\\ & = \frac{1}{b-a}\frac{1}{(x+a)^{n}(x+b)^{m-1}} + \frac{1}{b-a}\frac{1}{(x+a)^{n-1}(x+b)^{m}} \end{align*}

Perhaps there is some recurrence relation to find out the result.

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