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I was trying to see if there is a simple way to compute the following integral, where $0, $$ \int_a^b\log(x)\frac{\sqrt{(x-a)(b-x)}}{x(1-x)}dx. $$ Any idea ?

NB : With change of variables and the use of series expansion of $\log(1+x)$, one can reduce the problem to compute for all $k\geq 0$ $$ \int_0^{b-a}y^k\frac{\sqrt{y(b-a-y)}}{(y+a)(1-a-y)}dy, $$ but then I'm stuck ...

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    integration by parts may help since $log'(x) = 1/x$.2012-03-12
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    @Anon : Of course, that's why it is assumed $0.2012-03-13
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    @Quartz : And do you know a primitive of the other part ?2012-03-13

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