> 
> $$
> \mathbf{\mbox{Evaluate:}}\qquad
> \int\_{0}^{5} \frac{\pi(1+\frac{1}{2\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}} 
> \,\,\mathrm{d}x
> $$
> 
> 
> 

This is a very ugly integral, but appears to have a very simple closed form of: $$\Gamma(\frac15)\Gamma(\frac45)$$ Mathematica can evaluate this integral, but WolframAlpha doesn't even give a correct numerical answer. I have tried many techniques on this integral but have not been able to crack it at all.

Any help on this integral would be greatly appreciated. Thank you!