<s>[INST] Antiderivative of $ (x^2 + c)^{-3/2} $ What method should be used to determine the antiderivative of this expression?
Edit: I have $ c > 0 $ in the problem I'm working on.
 [/INST] $$\int(x^2+c)^{-3/2}dx=\int\frac{1}{\sqrt{(x^2+c)^3}}$$
Let us use a new variable $t$ such that $\sqrt{x^2+c}=x+t$. This is the first Euler substitution.
From this formula we can solve for $x$ to get $x=\frac{c-t^2}{2t}$, $dx=\frac{-c-t^2}{2t^2}dt$, $\sqrt{x^2+c}=x+t=\frac{c+t^2}{2t}$, and $\sqrt{(x^2+c)^3}=\frac{(c+t^2)^3}{8t^3}$.
We put these into the integral and get
$$\int\frac{8t^3}{(c+t^2)^3}\frac{-(c+t^2)}{2t^2}dt=-2\int\frac{2t}{(c+t^2)^2}dt=2\frac{1}{c+t^2}$$
We can return to the variable $x$ since $t=\sqrt{x^2+c}-x$. Therefore 
$$\int(x^2+c)^{-3/2}=\frac{2}{c+\left(\sqrt{x^2+c}-x\right)^2}$$
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