Indefinite integral of $\frac{\sqrt{1+x}}{\sqrt{1+x^a}}$

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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.