Integral $\int_{0}^{\infty}\frac1{\sqrt[\alpha]{1+x^\beta}}dx$ Is there a general answer for the integral of the form:$$\int_{0}^{\infty}\dfrac1{\sqrt[\alpha]{1+x^\beta}}dx$$

Using Mathematica:
Assuming[{1 > \[Alpha] > 0, \[Beta] > 1}, 
 Integrate[1/(1 + x^\[Beta])^(1/\[Alpha]), {x, 0, \[Infinity]}]]

$\dfrac{\Gamma \left(1+\frac{1}{\beta }\right) \Gamma \left(\frac{1}{\alpha
   }-\frac{1}{\beta }\right)}{\Gamma \left(\frac{1}{\alpha }\right)}$
