I am reading about elliptic integrals, and there is an article showing how to transform general form $\int R(x, \sqrt{P(x)})\,\mathrm{d}x$ with $P(x)$ being a polynomial of degree 3 or 4. It boils down to the three type of elliptic integrals (ie, non-elementary):
$$ \int \frac{1}{\sqrt{P(x)}}\,\mathrm{d}x\tag{$I_0$} $$
$$ \int \frac{x^2}{\sqrt{P(x)}}\,\mathrm{d}x\tag{$I_2$} $$ and $$ \int \frac{1}{(x-b)\sqrt{P(x)}}\,\mathrm{d}x\tag{$H_1$} $$
It is explicitly mentioned that the $I_1$ integral, $\int \frac{x}{\sqrt{P(x)}}\,\mathrm{d}x$, is elementary, but didn't show how to integrate it. I searched the web but found nothing similar. Could someone give a hint how to reduce it to elementary integral?
TIA