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For a thermodynamics project I'm working on, I need to evaluate this integral:

$\int \frac{(a-bx)(x-c)^d}{x^3}dx$, where $a,b,c,$ and $d$ are all positive constants.

I tried evaluating it on Wolfram Alpha, but it's giving a solution based on the hypergeometric series. Is there any other way of evaluating this integral? Or are there any good approximations of this integral in terms of elementary functions of $x$? I'm not looking for numerical solutions, but rather analytic solutions/approximations.

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  • $\begingroup$ You can make a taylor expansion and integrate it. $\endgroup$ – Ragnar Dec 26 '13 at 15:36
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How about using the binomial theorem?

$$(x-c)^d=\sum_{i=0}^{d}\binom{d}{i}x^{d-i}(-c)^i.$$

Then, you'll have $$\int\sum_{i=0}^{d}a\binom{d}{i}x^{d-3-i}(-c)^idx-\int\sum_{i=0}^{d}b\binom{d}{i}x^{d-2-i}(-c)^idx.$$

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Expand $ (x-c)^d $ using Binomial Theorem and the function will simplify to a finite sum of powers of x, which (given you remember to treat the $1\over x$ case separately) is trivial to integrate.

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