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]

-32/5*(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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Rubi [A]  time = 0.37, antiderivative size = 190, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.10, 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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fricas [A]  time = 0.47, size = 76, normalized size = 0.40 \[ \frac {16}{765765} \, {\left ({\left (231 \, \sqrt {x} - 1304\right )} \sqrt {\sqrt {x} + 1} + {\left ({\left (3003 \, \sqrt {x} - 4672\right )} \sqrt {\sqrt {x} + 1} - 3528 \, \sqrt {x} + 8752\right )} \sqrt {\sqrt {\sqrt {x} + 1} + 1} + 45045 \, x + 4613 \, \sqrt {x} - 28152\right )} \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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maple [A]  time = 0.01, size = 121, normalized size = 0.64 \[ -\frac {32 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {5}{2}}}{5}+\frac {48 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {7}{2}}}{7}+\frac {112 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {9}{2}}}{9}-\frac {320 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {11}{2}}}{11}+\frac {288 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {13}{2}}}{13}-\frac {112 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {15}{2}}}{15}+\frac {16 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.92, size = 120, normalized size = 0.63 \[ \frac {16}{17} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {112}{15} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {288}{13} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {320}{11} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {11}{2}} + \frac {112}{9} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {9}{2}} + \frac {48}{7} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {32}{5} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1}\, dx \]

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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