3.717 \(\int \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{x}}}} \, dx\)

Optimal. Leaf size=190 \[ \frac{16}{17} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{17/2}-\frac{112}{15} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{15/2}+\frac{288}{13} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{13/2}-\frac{320}{11} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{11/2}+\frac{112}{9} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{9/2}+\frac{48}{7} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{7/2}-\frac{32}{5} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{5/2} \]

[Out]

(-32*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(5/2))/5 + (48*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(7/2))/7 + (112*(1 + S
qrt[1 + Sqrt[1 + Sqrt[x]]])^(9/2))/9 - (320*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(11/2))/11 + (288*(1 + Sqrt[1 +
Sqrt[1 + Sqrt[x]]])^(13/2))/13 - (112*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(15/2))/15 + (16*(1 + Sqrt[1 + Sqrt[1
+ Sqrt[x]]])^(17/2))/17

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Rubi [A]  time = 0.366168, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1618, 1620} \[ \frac{16}{17} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{17/2}-\frac{112}{15} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{15/2}+\frac{288}{13} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{13/2}-\frac{320}{11} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{11/2}+\frac{112}{9} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{9/2}+\frac{48}{7} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{7/2}-\frac{32}{5} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]],x]

[Out]

(-32*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(5/2))/5 + (48*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(7/2))/7 + (112*(1 + S
qrt[1 + Sqrt[1 + Sqrt[x]]])^(9/2))/9 - (320*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(11/2))/11 + (288*(1 + Sqrt[1 +
Sqrt[1 + Sqrt[x]]])^(13/2))/13 - (112*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(15/2))/15 + (16*(1 + Sqrt[1 + Sqrt[1
+ Sqrt[x]]])^(17/2))/17

Rule 1618

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[PolynomialQuotient[Px, a + b
*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n, x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[PolynomialRema
inder[Px, a + b*x, x], 0]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{x}}}} \, dx &=2 \operatorname{Subst}\left (\int x \sqrt{1+\sqrt{1+\sqrt{1+x}}} \, dx,x,\sqrt{x}\right )\\ &=4 \operatorname{Subst}\left (\int x \left (-1+x^2\right ) \sqrt{1+\sqrt{1+x}} \, dx,x,\sqrt{1+\sqrt{x}}\right )\\ &=8 \operatorname{Subst}\left (\int x^3 \sqrt{1+x} \left (-2+x^2\right ) \left (-1+x^2\right ) \, dx,x,\sqrt{1+\sqrt{1+\sqrt{x}}}\right )\\ &=8 \operatorname{Subst}\left (\int x^3 (1+x)^{3/2} \left (2-2 x-x^2+x^3\right ) \, dx,x,\sqrt{1+\sqrt{1+\sqrt{x}}}\right )\\ &=8 \operatorname{Subst}\left (\int \left (-2 (1+x)^{3/2}+3 (1+x)^{5/2}+7 (1+x)^{7/2}-20 (1+x)^{9/2}+18 (1+x)^{11/2}-7 (1+x)^{13/2}+(1+x)^{15/2}\right ) \, dx,x,\sqrt{1+\sqrt{1+\sqrt{x}}}\right )\\ &=-\frac{32}{5} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{5/2}+\frac{48}{7} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{7/2}+\frac{112}{9} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{9/2}-\frac{320}{11} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{11/2}+\frac{288}{13} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{13/2}-\frac{112}{15} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{15/2}+\frac{16}{17} \left (1+\sqrt{1+\sqrt{1+\sqrt{x}}}\right )^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.103578, size = 135, normalized size = 0.71 \[ \frac{16 \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{5/2} \left (231 \sqrt{x} \left (-377 \sqrt{\sqrt{\sqrt{x}+1}+1}+195 \sqrt{\sqrt{x}+1}+365\right )+8 \left (252 \sqrt{\sqrt{x}+1} \sqrt{\sqrt{\sqrt{x}+1}+1}+8642 \sqrt{\sqrt{\sqrt{x}+1}+1}-4865 \sqrt{\sqrt{x}+1}-8221\right )\right )}{765765} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]],x]

[Out]

(16*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(5/2)*(8*(-8221 + 8642*Sqrt[1 + Sqrt[1 + Sqrt[x]]] - 4865*Sqrt[1 + Sqrt[
x]] + 252*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqrt[x]]) + 231*(365 - 377*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 195*Sq
rt[1 + Sqrt[x]])*Sqrt[x]))/765765

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Maple [A]  time = 0.013, size = 121, normalized size = 0.6 \begin{align*} -{\frac{32}{5} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{5}{2}}}}+{\frac{48}{7} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{7}{2}}}}+{\frac{112}{9} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{9}{2}}}}-{\frac{320}{11} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{11}{2}}}}+{\frac{288}{13} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{13}{2}}}}-{\frac{112}{15} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{15}{2}}}}+{\frac{16}{17} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x)

[Out]

-32/5*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(5/2)+48/7*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(7/2)+112/9*(1+(1+(1+x^(1/2))
^(1/2))^(1/2))^(9/2)-320/11*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(11/2)+288/13*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(13/
2)-112/15*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(15/2)+16/17*(1+(1+(1+x^(1/2))^(1/2))^(1/2))^(17/2)

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Maxima [A]  time = 1.00248, size = 162, normalized size = 0.85 \begin{align*} \frac{16}{17} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{17}{2}} - \frac{112}{15} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{15}{2}} + \frac{288}{13} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{13}{2}} - \frac{320}{11} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{11}{2}} + \frac{112}{9} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{9}{2}} + \frac{48}{7} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{7}{2}} - \frac{32}{5} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

16/17*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(17/2) - 112/15*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(15/2) + 288/13*(sqr
t(sqrt(sqrt(x) + 1) + 1) + 1)^(13/2) - 320/11*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(11/2) + 112/9*(sqrt(sqrt(sqrt
(x) + 1) + 1) + 1)^(9/2) + 48/7*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(7/2) - 32/5*(sqrt(sqrt(sqrt(x) + 1) + 1) +
1)^(5/2)

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Fricas [A]  time = 1.54633, size = 293, normalized size = 1.54 \begin{align*} \frac{16}{765765} \,{\left ({\left (231 \, \sqrt{x} - 1304\right )} \sqrt{\sqrt{x} + 1} +{\left ({\left (3003 \, \sqrt{x} - 4672\right )} \sqrt{\sqrt{x} + 1} - 3528 \, \sqrt{x} + 8752\right )} \sqrt{\sqrt{\sqrt{x} + 1} + 1} + 45045 \, x + 4613 \, \sqrt{x} - 28152\right )} \sqrt{\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

16/765765*((231*sqrt(x) - 1304)*sqrt(sqrt(x) + 1) + ((3003*sqrt(x) - 4672)*sqrt(sqrt(x) + 1) - 3528*sqrt(x) +
8752)*sqrt(sqrt(sqrt(x) + 1) + 1) + 45045*x + 4613*sqrt(x) - 28152)*sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(1+(1+x**(1/2))**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out