Evaluate the integral $\int_0^\infty\dfrac{x}{\sqrt{x^5+1}}dx$

Question

$$\int_0^\infty\dfrac{x}{\sqrt{x^5+1}}dx$$

I tried out few things, but none helped at all. Is it possible to evaluate the integral using Euler's substitutions (as demanded by my professor)? Also is there any other elementary way of solving it?

Answer 1

Hint:

$$B(x,y)=\int_0^{\infty}\frac{t^{x-1}}{(1+t)^{x+y}}dt$$

Where $B(x,y)$ is the Standard Beta function and $\displaystyle B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ where $\Gamma(z)$ is the Complete Gamma function.

In your given question, what happens if you substitute $\displaystyle x=t^{\frac{1}{5}}$?