Indefinite integral: chance of analytic solution?

Question

Is there a chance the following indefinte integral has an analytic solution?

$I(a, b) = \int \mathrm{d}x \sqrt{a^x +b^x}$

My attempts and those of Mathematica proved fruitless thus far...

Any suggestions for methods of attack are most welcome.

Answer 1

Let $\alpha = 2\frac{\log(b/a)}{\log a}$. Then Using the substitution $t = a^{x/2}$ we find that

$$ I(a, b) = \frac{2}{\log a} \int \sqrt{1 + t^{\alpha}} \, dt. $$

I highly doubt that we have a general closed form for this. For example, this reduces to an elliptic integral when $\alpha = 3$, which is obviously non-elementary.