How to perform this integration in terms of elementary functions?

Question

I need to perform this in tegration in terms of elementary functions... I'm not used to perform integrations like this, so a detailed answer would be very welcome. My integral is the following, % $$ \int \frac{dx}{\sqrt{P(x)}} \,\, , $$

with

$$ P(x) = \frac{1}{27}-x^{2}(1+2x) \,\, , \qquad x \in \mathbb{R}\,\, . $$ Once the roots of $P(x)$ are $r_{1}=-1/6$ and the following 2-algebric multiplicity $r_{2}=r_{3} = 1/3$, we can Put the polynomial in the form,

$$ P(x) = 2\left(x +\frac{1}{6}\right)\left(x-\frac{1}{3}\right)^{2} \,\, . $$

I'm stuck here.

Answer 1

Use the substitution $$x+\frac {1}{6}=t^2$$

For integrals of the form, $$\int \frac{1}{(ax+b) \sqrt{px+q}} \rm dx$$

A standard substitution is $px+q=t^2$

Answer 2

The hint:

Let $$\sqrt{x+\frac{1}{6}}=t.$$