Integrating $\int \sqrt{x+\sqrt{x+\sqrt{x}}}\, dx$

Question

How to evaluate $$I=\int \sqrt{x+\sqrt{x+\sqrt{x}}}\,dx$$ I am trying this question by substituting $x$ $=$ $(\tan\theta)^{4}$. Now $dx$ $=$ $4(\tan\theta)^{3}(\sec\theta)^{2}d\theta$. So, finally the integral will become $$\int [4(\tan\theta)^{3}((\tan\theta)^{4}+((\tan\theta)^{4}+ (\tan\theta)^{2})^{1/2}))^{1/2}(\sec\theta)^{2}]d\theta.$$ But I can't proceed further. Please help me out.

Answer 1

Hoping that I properly read it, you want to compute $$I=\int \sqrt{x+\sqrt{x+\sqrt{x}}}\,dx$$

I think that, beside numerical integration, the only possibility is a series expansion around $x=0$

$$\sqrt{x+\sqrt{x+\sqrt{x}}}=$$ $$x^{1/8}+\frac{x^{5/8}}{4}+\frac{x^{7/8}}{2}-\frac{3 x^{9/8}}{32}-\frac{x^{11/8}}{8}-\frac{9 x^{13/8}}{128}+\frac{5 x^{15/8}}{64}+O\left(x^{17/8}\right)$$

Integrating term wise between $0$ and $1$, the above very truncated series gives $$\frac{590445223}{486748080}=1.213$$ while numerical integration gives $1.219$.

This is not fantastic, for sure !

Answer 2

Using: $$\sqrt{x+\sqrt{x+\sqrt{x}}}=\sqrt{x} \sqrt{1+\sqrt{\frac{1}{x}+\frac{1}{x^{3/2}}}}$$ and using substituting: $x^{-3/2}=t^3$,$t+1=u$,we have simpler form to integrate by CAS:$$\int -\frac{2 \sqrt{u^{3/2}+1-\sqrt{u}}}{(u-1)^4} \, du$$ and with help of Mathematica gives long output,solution by: Elliptic integral of the first kind, complete Elliptic integral of the first kind and complete Elliptic integral of the third kind.

Solution with Mathematica ,the code:

S = Integrate[-2 Sqrt[u^(3/2) + 1 - Sqrt[u]]/(u - 1)^4, u] /. u -> t + 1 /. t -> 1/Sqrt[x] // Simplify(* -(1/( 4 Sqrt[1 + Sqrt[1 + 1/Sqrt[x]]/Sqrt[ x]])) (-(1/ 3) (Sqrt[1 + 1/Sqrt[x]] + Sqrt[x]) (-3 + 3 Sqrt[1 + 1/Sqrt[x]] + 2 Sqrt[1 + 1/Sqrt[x]] Sqrt[x] + 8 x) + Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[1 - # + #^3& , 1, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] (1/Sqrt[( Sqrt[1 + 1/Sqrt[x]] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])] EllipticF[ ArcSin[Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[ 1 - # + #^3& , 3, 0])]], ( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] (Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 3, 0]) Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 2, 0])/( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])] + 1/Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[1 - # + #^3& , 2, 0])/( Root[1 - # + #^3& , 2, 0] - Root[ 1 - # + #^3& , 3, 0])] (Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 2, 0]) (-EllipticF[ ArcSin[Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[1 - # + #^3& , 3, 0])]], ( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] Root[1 - # + #^3& , 1, 0] + EllipticE[ ArcSin[Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[1 - # + #^3& , 3, 0])]], ( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] (Root[1 - # + #^3& , 1, 0] - Root[1 - # + #^3& , 3, 0])) Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[1 - # + #^3& , 3, 0])] + 1/(-1 + Root[1 - # + #^3& , 3, 0]) 2 EllipticPi[(-Root[1 - # + #^3& , 2, 0] + Root[ 1 - # + #^3& , 3, 0])/(-1 + Root[1 - # + #^3& , 3, 0]), ArcSin[Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[ 1 - # + #^3& , 3, 0])]], ( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] Sqrt[-(((Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 2, 0]) (Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 3, 0]))/(Root[1 - # + #^3& , 2, 0] - Root[ 1 - # + #^3& , 3, 0])^2)] (-Root[1 - # + #^3& , 2, 0] + Root[1 - # + #^3& , 3, 0]) + 1/(1 + Root[1 - # + #^3& , 3, 0]) 2 EllipticPi[(-Root[1 - # + #^3& , 2, 0] + Root[ 1 - # + #^3& , 3, 0])/(1 + Root[1 - # + #^3& , 3, 0]), ArcSin[Sqrt[(-Sqrt[1 + 1/Sqrt[x]] + Root[ 1 - # + #^3& , 3, 0])/(-Root[1 - # + #^3& , 2, 0] + Root[ 1 - # + #^3& , 3, 0])]], ( Root[1 - # + #^3& , 2, 0] - Root[1 - # + #^3& , 3, 0])/( Root[1 - # + #^3& , 1, 0] - Root[ 1 - # + #^3& , 3, 0])] Sqrt[-(((Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 2, 0]) (Sqrt[1 + 1/Sqrt[x]] - Root[ 1 - # + #^3& , 3, 0]))/(Root[1 - # + #^3& , 2, 0] - Root[ 1 - # + #^3& , 3, 0])^2)] (-Root[1 - # + #^3& , 2, 0] + Root[1 - # + #^3& , 3, 0]))) *) Plot[{Evaluate[D[S, x]], Sqrt[x + Sqrt[x + Sqrt[x]]]}, {x, 0, 10}, PlotStyle -> {Red, {Dashed, Black}}]