Irrational Integral

Question

I've tried in many ways to compute this integral but I'm not able to find any solution. Even Wolfram can not compute this. So my question is: Is that even possible to compute? $$\int { \frac { dx }{ 1+\sqrt { x } +\sqrt { x+1 } +\sqrt { x+2 } } }$$

Answer 1

Hint:

Let $x=u-1$ ,

Then $\int\dfrac{dx}{1+\sqrt x+\sqrt{x+1}+\sqrt{x+2}}$

$=\int\dfrac{du}{1+\sqrt u+\sqrt{u-1}+\sqrt{u+1}}$

$=\int\dfrac{\sqrt{u-1}+\sqrt{u+1}-1-\sqrt u}{(1+\sqrt u+\sqrt{u-1}+\sqrt{u+1})(\sqrt{u-1}+\sqrt{u+1}-1-\sqrt u)}~du$

$=\int\dfrac{\sqrt{u-1}+\sqrt{u+1}-1-\sqrt u}{(\sqrt{u-1}+\sqrt{u+1})^2-(1+\sqrt u)^2}~du$

$=\int\dfrac{\sqrt{u-1}+\sqrt{u+1}-1-\sqrt u}{u-1-2\sqrt u+2\sqrt{u^2-1}}~du$