What substitution should i make for this integral

Question

$$ \int_{0}^{2} \sqrt{x+ \sqrt{\frac{x^2}{2}+\sqrt{\frac{x^3}{3}+1}}}dx\> $$

Wolfram alpha gives its answer as 3.01376.

Can any one provide me a solution for this?

Answer 1

Too long for a comment: Imagine the much simpler integrand $\sqrt{x+\sqrt{x^2+1}}~.~$

A trivial trigonometric or hyperbolic substitution of the form $x=\tan t$ or $x=\sinh u$

yields a closed form expression for its primitive, in terms of elementary functions.

By applying what seems to be only a slight modification, and changing its form to

$\sqrt{x+\sqrt{\dfrac{x^2}2+1}}$, the result becomes inexpressible without the aid of hypergeometric

series, or incomplete beta functions, or elliptic integrals : take your pick. The expression

in question, following the hyperbolic substitution $x=\sqrt2~\sinh t$, can be found here.

$($Trigonometric substitutions yield even uglier results$)$. Replacing $\bf1$ by $\sqrt{\dfrac{x^3}3+1}$ ,

even these special functions are unable to help us express the anti-derivative. For more

information on this topic, see Liouville's theorem and the Risch algorithm.