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I have tried to find $\int{\biggl(\dfrac{\sqrt{x+1}}{x-1}\biggr)^x}dx$ but I don't know how to do it, because it combines $u^x$ and $\dfrac{u}{v}$.

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    @Garmen1778 there are some functions which ,(they say) cannot be integrated (without mathematica), this one from your imagination might belong to that set of questions.2012-05-08

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You can do... $\int f(x)^x\; dx=\int e^{x\ln f(x)}\; dx=\int e^{\alpha(x)}\; dx$ where $f(x)=\frac{\sqrt{x+1}}{x-1}$

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    The cuestion about natural o decimal logarithm.. **Look** the exponential!!! (Edited notation)2012-05-09