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Please give me some hints and clues.

Thank you

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The book you're thinking of is Calculus: A Differentiable Approach, Third Edition, by James Stewart and Thomas H. Malthouse.

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Please give me some hints and clues.

Thank you

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Elliptic integrals!

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[tex]x^3+x^2+x+1 \ge 0[/tex]

[tex]x \ge -1[/tex]

You should take care of integral interval for finite and real result.

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It's a pretty interesting subject, elliptic integrals and functions if you're into that sort of thing. Check Wikipedia article: Elliptic integralsaskor said:How do you integrate ##\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx##?

Please give me some hints and clues.

. . . , 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.

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

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

How do you integrate above?

Please give me a clues and hints.

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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.

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