I'm a physics undergrad who recently came across the following integral in one of their problems:
$$\int_0^{\infty}\frac{1}{(x+x^3)^{\frac{1}{2}}}\approx 3.708$$
I was curious as to how this was evaluated so went to Wolfram where it solved as
$$\int_0^{\infty}\frac{1}{(x+x^3)^{\frac{1}{2}}} =\frac{8 \cdot\Gamma(\frac{5}{4})}{\sqrt{\pi}}$$
I am slightly familiar with the Gamma integral, though I've only ever really seen it in passing. I was wondering if a) someone could show me how one arrives at the above expression for the integral, and b) if I could be pointed towards any resources which show one how to tackle such integrals (particularly for physics students who may not be used to the same level of mathematical rigour).
For those interested, the integral came up when considering the maximum comoving co-ordinate of a particle with finite velocity in the flat (Einstein-deSitter) universe.