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

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 + Sqr
t[x]]])^(7/2))/7 + (112*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(9/2))/9 - (320*(1 + S
qrt[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 = 0.557395, 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 \[ \frac{16}{17} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{17/2}-\frac{112}{15} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{15/2}+\frac{288}{13} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{13/2}-\frac{320}{11} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{11/2}+\frac{112}{9} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{9/2}+\frac{48}{7} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{7/2}-\frac{32}{5} \left (\sqrt{\sqrt{\sqrt{x}+1}+1}+1\right )^{5/2} \]

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 + Sqr
t[x]]])^(7/2))/7 + (112*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(9/2))/9 - (320*(1 + S
qrt[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 in Sympy [A]  time = 18.1077, size = 165, normalized size = 0.87 \[ \frac{16 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{17}{2}}}{17} - \frac{112 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{15}{2}}}{15} + \frac{288 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{13}{2}}}{13} - \frac{320 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{11}{2}}}{11} + \frac{112 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{9}{2}}}{9} + \frac{48 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{7}{2}}}{7} - \frac{32 \left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+(1+(1+x**(1/2))**(1/2))**(1/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.128768, 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 + S
qrt[x]]] - 4865*Sqrt[1 + Sqrt[x]] + 252*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqr
t[x]]) + 231*(365 - 377*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 195*Sqrt[1 + Sqrt[x]])*Sqr
t[x]))/765765

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Maple [A]  time = 0.019, size = 121, normalized size = 0.6 \[ -{\frac{32}{5} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{5}{2}}}}+{\frac{48}{7} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{7}{2}}}}+{\frac{112}{9} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{9}{2}}}}-{\frac{320}{11} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{11}{2}}}}+{\frac{288}{13} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{13}{2}}}}-{\frac{112}{15} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{15}{2}}}}+{\frac{16}{17} \left ( 1+\sqrt{1+\sqrt{1+\sqrt{x}}} \right ) ^{{\frac{17}{2}}}} \]

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 = 0.739546, size = 162, normalized size = 0.85 \[ \frac{16}{17} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{17}{2}} - \frac{112}{15} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{15}{2}} + \frac{288}{13} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{13}{2}} - \frac{320}{11} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{11}{2}} + \frac{112}{9} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{9}{2}} + \frac{48}{7} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{7}{2}} - \frac{32}{5} \,{\left (\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1),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*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(13/2) - 320/11*(sqr
t(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 = 0.270573, size = 103, normalized size = 0.54 \[ \frac{16}{765765} \,{\left ({\left (231 \, \sqrt{x} - 1304\right )} \sqrt{\sqrt{x} + 1} +{\left ({\left (3003 \, \sqrt{x} - 4672\right )} \sqrt{\sqrt{x} + 1} - 3528 \, \sqrt{x} + 8752\right )} \sqrt{\sqrt{\sqrt{x} + 1} + 1} + 45045 \, x + 4613 \, \sqrt{x} - 28152\right )} \sqrt{\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1),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. \[ \int \sqrt{\sqrt{\sqrt{\sqrt{x} + 1} + 1} + 1}\, dx \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1),x, algorithm="giac")

[Out]

Exception raised: TypeError