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I have a problem with integration of following function:

 Integrate[
   x Log[(A Sqrt[x] + B + Sqrt[x - p^2])/(A Sqrt[x] + B - Sqrt[x - p^2])]/
     ((x + C)^2 (2 x + m Sqrt[x])),
   {x, 0, Infinity}]

Although I know that a result exists, I cannot compute it. After waiting few minutes, nothing happens. The result is exactly the same as the function in integral.

Could you give me some advice how I can solve this problem? Is there any way to force Mathematica to give the result?

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    $\begingroup$ Are there any constrains to the variables? Note that C is a built in symbol, and should preferably not be used. $\endgroup$ – Feyre Aug 11 '16 at 14:15
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    $\begingroup$ Mathematica returns expressions unevaluated when it doesn't know how to handle them. Why are you certain that a symbolic, closed-form result exists for this integral? It has five free variables, which makes me doubt that any closed-form result exists. $\endgroup$ – m_goldberg Aug 11 '16 at 14:34
  • $\begingroup$ If I'm not mistaken, the piece of the integrand involving the natural logarithm can be rewritten as $\log ( ... ) = 2 \tanh^{-1}[\sqrt{x - p^2}/(A \sqrt{x} + B)]$. This will be imaginary for $x < p^2$ (assuming real $A$ and $B$). Is this what you expect? $\endgroup$ – Michael Seifert Aug 11 '16 at 17:46
  • $\begingroup$ I know that because I am repeating calculations after my supervisor. All variables different than x have a physical meaning and values. $\endgroup$ – filipmor Aug 12 '16 at 6:54
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I think the problem is arising from Sqrt[x - p^2]. If we assume that a closed-form result exists, then we should be able to get a numerical result for any parameter values. If I try

Block[{a = 1, b = 1, c = 1, p = 1, m = 1}, 
 NIntegrate[x Log[(a Sqrt[x] + b + Sqrt[x - p^2])/(a Sqrt[x] + b - 
    Sqrt[x - p^2])]/((x + c)^2 (2 x + m Sqrt[x]))
, {x, p, Infinity}]]

0.257783

If I choose the limit from 0

Block[{a = 1, b = 1, c = 1, p = 1, m = 1}, 
 NIntegrate[x Log[(a Sqrt[x] + b + Sqrt[x - p^2])/(a Sqrt[x] + b - 
    Sqrt[x - p^2])]/((x + c)^2 (2 x + m Sqrt[x]))
, {x, 0, Infinity}]]

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>

0.257783 + 0.10605 I

When you try to evaluate it symbolically, it would be messier, and you would be stuck forever.

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