Recall that indefinite integral follows the formula: `int f(x) dx = F(x) +C`

where: `f(x)` as the integrand

`F(x)` as the anti-derivative function

`C` as the arbitrary constant known as constant of integration

For the given problem `int 1/(x^2+5)^(3/2)dx` , it resembles one of the formula from integration table. We may apply the integral formula for rational function with roots as:

`int 1/(u^2+a^2)^(3/2)du= u/(a^2sqrt(u^2+a^2))+C`

By comparing "`u^2+a^2` " with "`x^2+5` " , we determine the corresponding values as:

`u^2=x^2` then `u = x` and `du = dx`

`a^2 =5` then `a = sqrt(5)` .

Plug-in the corresponding values on the aforementioned integral formula for rational function with roots, we get:

`int 1/(x^2+5)^(3/2)dx =x/(5sqrt(x^2+5))+C`