How to integrate this $\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}} $ without trigonometric substitution? I have been looking for a possible solution but they are with trigonometric integration..
I need a solution for this function without trigonometric integration
$$\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}}$$

Let $I=\int \frac{dx}{\sqrt{x^2+4}} \,;\, J=\int \frac{dx}{\sqrt{x^2+4}^3}$
We integrate $I$ by parts, we get:
$u=(x^2+4)^{-1/2}, dv=dx$
$du=-x(x^2+4)^{-3/2}, v=x$
Thus
$$I= \frac{x}{\sqrt{x^2+4}}+ \int \frac{x^2}{\sqrt{x^2+4}^3}= \frac{x}{\sqrt{x^2+4}}+ \int \frac{x^2+4-4}{\sqrt{x^2+4}^3}=$$
$$= \frac{x}{\sqrt{x^2+4}}+ I-4J=$$
Thus, canceling the $I$ we get
$$4J=  \frac{x}{\sqrt{x^2+4}} +C$$
