How Do You Integrate 1/√(x^3 + x^2 + x + 1) dx?

[askor]

How do you integrate $\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx$?

Please give me some hints and clues.

Thank you

 

[fresh_42]

I would write the polynomial as $(x^2+1)(x+1)$ and try a suitable substitution like $x^2+1=u$ or similar.

 

[Office_Shredder]

Do you have specific bounds to integrate between? Wolfram alpha suggests the answer to this is you do not.

 

[martinbn]

Elliptic integrals!

 

[anuttarasammyak]

For
$$x^3+x^2+x+1 \ge 0$$
$$x \ge -1$$
You should take care of integral interval for finite and real result.

 

[aheight]

How do you integrate $\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx$?

Please give me some hints and clues.

It's a pretty interesting subject, elliptic integrals and functions if you're into that sort of thing. Check Wikipedia article: Elliptic integrals

. . . , with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

So it looks like you can express your integand as rational functions and the first, second, and third elliptical integrals and compute them using arithemetic-geometric means as per the reference. Sounds like an interesting research project but looks like it would take a bit of effort.

 

[askor]

So, it's an advance integration that did not taught in standard Calculus textbook, am I right?

May I know what book that teach an integration like this?

 

[askor]

What about this?

$$\int \frac{1}{\sqrt{x^3 + 6x^2 + 11x + 6}} \, dx$$

How do you integrate above?

Please give me a clues and hints.

 

[fresh_42]

I always test those integrals on WolframAlpha. You get a good impression of its complexity, if it doesn't find an answer, or it does and the result is complicated.

And here are some exhaustive lists:
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Bestimmte_Integrale
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Integrale

It's the wrong language, but who cares, the formulas are the same.