Evaluating indefinite integral with any hint or solution

Question

How can I evaluate this indefinite integral.

$$ \int\frac{dx}{x\sqrt{x^3+x+1}} $$

any hit would be appreciated.

Answer 1

This "reduces" to an incomplete elliptic integral. Somewhat more generally,

$$ \int \dfrac{dx}{x \sqrt{(x-a)(x-b)(x-c)}} = {\frac {\pm 2\,i}{a \sqrt {a-c}}{\it EllipticPi} \left( {\frac {\sqrt {a-x}}{\sqrt {a-b}}},{\frac {a-b}{a}},{\frac {\sqrt {a-b}}{\sqrt {a-c}}} \right) }$$

(using Maple's notation). In your case you want to take $a,b,c$ to be the three roots of $x^3+x+1$ (one real root approximately $-0.682327803828019$, two complex roots approximately $0.341163901914009693 \pm 1.16154139999725192\,i$).