Is there an elegant way to evaluate $ I={ \int \sqrt[8]{\frac{x+1}{x}} \ \mathrm{d}x}$? Is there an elegant way to evaluate the following integral?  

$$ I={ \int \sqrt[8]{\dfrac{x+1}{x}} \ \mathrm{d}x}$$   

This seems to me a very lengthy question, yet it was given in my weekly worksheet, so there must be an elegant solution.  
Any help will be appreciated.   

I have Mathematica 10.4 and what it returns for the primitive function is too lengthy! Off the top of my head, I don't think that there is any other way but the strategy to obtain the final answer is not that much complicated.! :) I just explain the path that you may go to obtain this result.

Firstly, we set 
$$\begin{align}
t &= \frac{x+1}{x} \\
dx &= \frac{1}{(t-1)^2} dt
\end{align}$$
and the integral will become
$$I=-\int \frac{\sqrt[8]{t}}{(t-1)^2} dt $$
and then setting 
$$\begin{align}
t &= u^8 \\
dt &= 8u^7du
\end{align}$$
will lead to
$$I=-8\int \frac{u^8}{(u^8-1)^2} du$$
and from here on, you should use partial fraction to reach what Mathematica gives!
