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$$\int_{x_1}^{x_2}\frac{\sqrt{\frac{1}{3}x^3+a}}{(1-x)\sqrt{x}\sqrt{-\frac{4}{3}x^3+x^2-a}}dx$$

where $a\in(0,\frac{1}{12})$ is a constant. In this case, $-\frac{4}{3}x^3+x^2-a=0$ has exactly two roots in $(0,1)$. $x_1,x_2$ are the two roots in $(0,1)$.

Thank you.

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    Where did you see this?2012-05-05
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    someone asked me this. he need to compute this in his paper about Riemannia Geometry.2012-05-05
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    if this integration cannot be computed, proving the integration increases as $a$ increases in also helpful2012-05-05
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    It seems to me it can, but it will be rather complicated, and the result might have to be expressed in terms of elliptic integrals. If you're fine with that...2012-05-05
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    The answer is $2\pi/3$, by wolframalpha.2012-05-05
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    @Yimin where is $a$?2012-05-05
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    sorry, I misunderstood your problem, apologize for that spam. it is a simple case that a=0.2012-05-05
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    @J.M. , er, any results will be fine. I would appreciate it if you post your results!2012-05-05

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