Help with $\int_{0}^{\infty}\frac{1}{\sqrt{x^{5}+1}}dx$ $$\int_{0}^{\infty}\frac{1}{\sqrt{x^{5}+1}}dx$$
So far,
$$\int\frac{1}{\sqrt{x^{5}+1}}dx=\int\frac{1}{\sqrt{x^{4}\left(x+\frac{4}{x^{4}}\right)}}dx=\int\frac{1}{x^{2}\sqrt{x+\frac{4}{x^{4}}}}dx$$
but from there I do not know how to proceed, I have tried several
changes of variable but I have not obtained anything clear. 
I only can work with real methods. Any suggestion? Thanks!
Edit: It's sufficiente to prove that it converges

The convergence is straightforward since $\frac{1}{\sqrt{x^5+1}}$ is positive and bounded by $\min\left(1,\frac{1}{x^{5/2}}\right)$, so the integral is positive and less than $\frac{5}{3}$. Of course we can state something fairly more accurate. By Euler's Beta function our integral equals
$$ \int_{0}^{+\infty}\frac{z^{-4/5}\,dz}{\sqrt{z+1}}=\frac{1}{5}\int_{1}^{+\infty}\frac{(z-1)^{-4/5}}{\sqrt{z}}\,dz = \frac{1}{5}\int_{0}^{1}z^{-7/10}(1-z)^{-4/5} dz=\color{red}{\frac{\Gamma(3/10)\Gamma(2/10)}{5\sqrt{\pi}}} $$
or, by exploiting the Weierstrass product for the $\Gamma$ function,
$$ \frac{5}{3}\prod_{n\geq 1}\left(1+\frac{3}{10m}\right)^{-1}\left(1+\frac{2}{10n}\right)^{-1}\left(1+\frac{1}{2n}\right)=\color{red}{\frac{5}{3}\prod_{n\geq 1}\left(1+\frac{3}{50n^2+25n}\right)^{-1}}. $$
By creative telescoping it turns out that the value of the integral is $\leq\frac{25}{16}=\left(\frac{5}{4}\right)^2$, with a very small absolute error.
