A horrid-looking integral $\int_{0}^{5} \frac{\pi(1+\frac{1}{2+\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}} $ 
$$
\mathbf{\mbox{Evaluate:}}\qquad
\int_{0}^{5}  \frac{\pi(1+\frac{1}{2\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}} 
\,\,\mathrm{d}x
$$

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!

$u=\sqrt{x}$, we have
$$
\int_{0}^{5}  \frac{\pi(1+\frac{1}{2\sqrt{x}} )}{\sqrt{10}\sqrt{\sqrt{x}+x}} 
\,\,\mathrm{d}x=\frac{\pi}{\sqrt{10}}\int_{0}^{\sqrt{5}}  \frac{2u+1} {\sqrt{u+u^2}}du=\frac{2\pi}{\sqrt{10}}\sqrt{u^2+u}\Big{|}_{0}^{\sqrt{5}}
$$
