One dimensional integration that Mathematica cannot do

[CAF123]

I want to evaluate $$\int_{a+b-c}^s\,\text{d}x\, \frac{(-x+ab/c)^{\epsilon}}{(x+c-a-b)^{\epsilon+1} (a-x)},$$ where $a,b,c,\epsilon$ are numbers, and to be treated as constants in the integration. I put this into mathematica and an hour later it is still attempting to evaluate it so I aborted the calculation. I was just wondering if someone could put the following code into Mathematica and see if it is also takes a long time on their machine?

Integrate[(-x + a*b/c)^e/(x + c - a - b)^(e + 1)/(a - x), {x,
a + b - c, s}, Assumptions -> e < 0 && b < 0 && a > 0]

I tried to progress with the intergral analytically:
Partial fractions gave a rewriting of the form $$\frac{1}{c-b}\int_{a+b-c}^{s} \text{d}x\, \frac{(-x+ab/c)^{\epsilon}}{(x+c-a-b)^{\epsilon}} \left(\frac{1}{x+c-a-b} + \frac{1}{a-x}\right)$$ $$ = \frac{1}{c-b}\left(\int_{a+b-c}^{s} \text{d}x\, \frac{(-x+ab/c)^{\epsilon}}{(x+c-a-b)^{\epsilon+1}} + \int_{a+b-c}^{s} \text{d}x\, \frac{(-x+ab/c)^{\epsilon}}{(x+c-a-b)^{\epsilon}} \frac{1}{a-x}\right)$$ but this probably did not help.

Thanks!

 

[Ssnow]

It is improper in $x=a+b-c$ ...

 

[CAF123]

I evaluated it initially without the factor of (a-x) in the denominator and an answer came out fine. But then I realized I had missed this piece upon typing it into Mathematica and now it takes a long time to give me an answer. $\epsilon < 0$ solves the fact it is improper at the boundary I think. Can you try putting it into your machine and seeing if is also takes a long time?

 

[Ssnow]

yes $\epsilon <0$ solve this, I think the problem is that when you divide by $\frac{1}{a-x}$ the computation become more complex..., I tried with Mathematica but Wolfram told me false Assumptions ...

 

[Ssnow]

You can try Taylor expanding the term $\frac{1}{a-x}$ but obviously is not an exact solution ...