Could $\int\frac{1}{x^{N+1}(x-1)}dx$ be solved analytically? I am trying to solve this integral: 
$$\int\frac{1}{x^{N+1}(x-1)}dx$$
I have tried integration by partial fraction, substitution and by parts. But, I can't solve it. So, I would like to ask could this  be solved?
Also, May I know when partial fraction does not exist? 
Thank you very much. 
Update:
N is any number that is greater than 0. Sorry for forgetting to include such information, and apology for any inconvenience caused. 

If you try a few integer values of $N$, you'll notice that partial fraction decomposition gives you a pattern that suggests
$$\frac{1}{x^{N+1}(x-1)} = \frac{1}{x-1} - \sum_{k=1}^{N+1} \frac{1}{x^k}$$ 
From there, simply integrate both sides. They're fairly trivial integrals when rewritten as so.
