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Are there any special functions or other techniques to solve integrals of the form:

$\int\frac{\sqrt{1+x}}{\sqrt{1+x^a}}\ dx$

where $a$ is a real number?

If the numerator were just 1, for instance, then the solution can be written with hypergeometric functions.

David G. Stork
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gigo318
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  • Even for $a=2$, the result is quite unwieldy and requires elliptic integrals. I wouldn't hold out too much hope for there being a very neat general, closed form solution. – Jam Jun 10 '20 at 17:47
  • $a=3$ or $a=-1$ are simple though. – zwim Jun 10 '20 at 17:52
  • $a=5$ elliptic integrals again. – GEdgar Jun 10 '20 at 17:54
  • Big difference between $\mathbb{Z}$ and $\mathbb{R}$. – David G. Stork Jun 10 '20 at 17:56
  • Specifying that $a$ is a real number was deliberate. I am aware that solutions can generally be found more easily for integer powers, but I am curious to see if there are any special functions that have been found that allow for solutions when there is an arbitrary real-valued (or if potentially rational) power under a radical. – gigo318 Jun 10 '20 at 18:31

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