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Is it possible to compute a formula for the following integral,

$$\int_{x_0}^{x_1} \sqrt{x^2-\frac{4a}{1-x}}\ dx, \ \ 0 < x_0 < x_1 < 1, \ 0 \leq a \leq \frac{1}{27}$$

The constraint over $a$ makes sure that the term inside the square root does not become negative and the constraint over $x_0$ and $x_1$ makes sure that the term $1-x$ stays positive.

I thought of making trigonometric substitution but couldn't figure out which one will simplify the integral. Other trivial algebraic substitutions doesn't seem to work too.

Appreciate any help in the form of hints or an answer. Thanks.

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  • $\begingroup$ It's not solvable in finite terms. It involves elliptic special functions $\endgroup$ – Raffaele Jul 13 '17 at 9:26
  • $\begingroup$ @Raffaele If it is not solvable in finite terms then is there a better representation of the integral. Possibly a representation which does not involve integration. And can you also point me to how one can numerically compute such integrals. Thanks! $\endgroup$ – Dhruv Kohli - expiTTp1z0 Jul 13 '17 at 10:43
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    $\begingroup$ @expiTTp1z0 The constraints on $x_0,x_1,a$ that you list are not sufficient to guarantee that the term inside the square-root is non-negative. For $0<a<\frac{1}{27}$, the cubic equation $x^3-x^2+4a=0$ has two distinct positive roots less than unity, $0<p<q<1$. The extra condition $0<p\le x_0<x_1\le q<1$ ensures that the term inside the square-root is non-negative. $\endgroup$ – David H Jul 29 '17 at 20:11

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