3.8.17 \(\int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx\) [717]

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

[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 = 0.25, 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 = {1632, 1634} \begin {gather*} \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} \end {gather*}

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 1632

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 1634

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 \text {Subst}\left (\int x \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {x}\right )\\ &=4 \text {Subst}\left (\int x \left (-1+x^2\right ) \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=8 \text {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 \text {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 \text {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 = 0.16, size = 168, normalized size = 0.88 \begin {gather*} \frac {16 \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \left (-8 \left (3519-1094 \sqrt {1+\sqrt {1+\sqrt {x}}}+163 \sqrt {1+\sqrt {x}}+584 \sqrt {1+\sqrt {1+\sqrt {x}}} \sqrt {1+\sqrt {x}}\right )+7 \left (659-504 \sqrt {1+\sqrt {1+\sqrt {x}}}+33 \sqrt {1+\sqrt {x}}+429 \sqrt {1+\sqrt {1+\sqrt {x}}} \sqrt {1+\sqrt {x}}\right ) \sqrt {x}+45045 x\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]]*(-8*(3519 - 1094*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 163*Sqrt[1 + Sqrt[x]]
+ 584*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqrt[x]]) + 7*(659 - 504*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 33*Sqrt[1 +
Sqrt[x]] + 429*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqrt[x]])*Sqrt[x] + 45045*x))/765765

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Maple [A]
time = 0.36, size = 121, normalized size = 0.64

method	result	size
derivativedivides	\(-\frac {32 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {5}{2}}}{5}+\frac {48 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {9}{2}}}{9}-\frac {320 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {288 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {16 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {17}{2}}}{17}\)	\(121\)
default	\(-\frac {32 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {5}{2}}}{5}+\frac {48 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {9}{2}}}{9}-\frac {320 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {288 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {16 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {17}{2}}}{17}\)	\(121\)

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,method=_RETURNVERBOSE)

[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 = 0.27, size = 120, normalized size = 0.63 \begin {gather*} \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 {gather*}

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 = 0.38, size = 76, normalized size = 0.40 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 7916 vs. \(2 (120) = 240\).
time = 35.37, size = 7916, normalized size = 41.66 \begin {gather*} \text {Too large to display} \end {gather*}

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]

16/765765*(7*(6435*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(17/2) - 58344*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(15/2) +
235620*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(13/2) - 556920*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(11/2) + 850850*(s
qrt(sqrt(sqrt(x) + 1) + 1) + 1)^(9/2) - 875160*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(7/2) + 612612*(sqrt(sqrt(sqr
t(x) + 1) + 1) + 1)^(5/2) - 291720*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(3/2) + 109395*sqrt(sqrt(sqrt(sqrt(x) + 1
) + 1) + 1))*sgn(70368744177664*(sqrt(sqrt(x) + 1) + 1)^92 - 6473924464345088*(sqrt(sqrt(x) + 1) + 1)^91 + 291
326600895528960*(sqrt(sqrt(x) + 1) + 1)^90 - 8545580292935516160*(sqrt(sqrt(x) + 1) + 1)^89 + 1837287624372765
32736*(sqrt(sqrt(x) + 1) + 1)^88 - 3086556782054646743040*(sqrt(sqrt(x) + 1) + 1)^87 + 42179809308639429132288
*(sqrt(sqrt(x) + 1) + 1)^86 - 481978846822841400164352*(sqrt(sqrt(x) + 1) + 1)^85 + 4697911198078384159588352*
(sqrt(sqrt(x) + 1) + 1)^84 - 39651330432185076620984320*(sqrt(sqrt(x) + 1) + 1)^83 + 2931836397160032337217454
08*(sqrt(sqrt(x) + 1) + 1)^82 - 1916656336440269370174734336*(sqrt(sqrt(x) + 1) + 1)^81 + 11160164453620451334
571425792*(sqrt(sqrt(x) + 1) + 1)^80 - 58223902019906429347317153792*(sqrt(sqrt(x) + 1) + 1)^79 + 273479024956
137655533112918016*(sqrt(sqrt(x) + 1) + 1)^78 - 1160956607882993155309408616448*(sqrt(sqrt(x) + 1) + 1)^77 + 4
467886822469532994953426239488*(sqrt(sqrt(x) + 1) + 1)^76 - 15624039803063454614788052615168*(sqrt(sqrt(x) + 1
) + 1)^75 + 49728771914087708805425247813632*(sqrt(sqrt(x) + 1) + 1)^74 - 144204022361387642459669217148928*(s
qrt(sqrt(x) + 1) + 1)^73 + 381099384933784007520636056371200*(sqrt(sqrt(x) + 1) + 1)^72 - 91748872521441581395
7123995336704*(sqrt(sqrt(x) + 1) + 1)^71 + 2009521130818998104990097239703552*(sqrt(sqrt(x) + 1) + 1)^70 - 399
4471142582563999654557691936768*(sqrt(sqrt(x) + 1) + 1)^69 + 7177812996901911023337169833951232*(sqrt(sqrt(x)
+ 1) + 1)^68 - 11588332903437268712897290291904512*(sqrt(sqrt(x) + 1) + 1)^67 + 166466900838183024506993560487
19872*(sqrt(sqrt(x) + 1) + 1)^66 - 20936686151898804312893580357140480*(sqrt(sqrt(x) + 1) + 1)^65 + 2238278803
8883899099152454346866688*(sqrt(sqrt(x) + 1) + 1)^64 - 19056354227487119677451342446592000*(sqrt(sqrt(x) + 1)
+ 1)^63 + 10446792239109173739071175649132544*(sqrt(sqrt(x) + 1) + 1)^62 + 1511753796217450360680303785148416*
(sqrt(sqrt(x) + 1) + 1)^61 - 12615704090193713988088537190236160*(sqrt(sqrt(x) + 1) + 1)^60 + 1821076901027652
4054435333169741824*(sqrt(sqrt(x) + 1) + 1)^59 - 15888618239925478635050328221810688*(sqrt(sqrt(x) + 1) + 1)^5
8 + 7264980298352403064896955164393472*(sqrt(sqrt(x) + 1) + 1)^57 + 2717159235634682624701237439758336*(sqrt(s
qrt(x) + 1) + 1)^56 - 8806173737385529153533018462224384*(sqrt(sqrt(x) + 1) + 1)^55 + 870458950951868157176149
6954765312*(sqrt(sqrt(x) + 1) + 1)^54 - 4141051044270206270604188407824384*(sqrt(sqrt(x) + 1) + 1)^53 - 944047
265435153343329682904317952*(sqrt(sqrt(x) + 1) + 1)^52 + 3441421759241742702311709805117440*(sqrt(sqrt(x) + 1)
+ 1)^51 - 2875820730830681791678590352359424*(sqrt(sqrt(x) + 1) + 1)^50 + 881068299799276284483428560142336*(
sqrt(sqrt(x) + 1) + 1)^49 + 656876670010853235917344051560448*(sqrt(sqrt(x) + 1) + 1)^48 - 9853147301419233940
87336160526336*(sqrt(sqrt(x) + 1) + 1)^47 + 512961170622589184570169885720576*(sqrt(sqrt(x) + 1) + 1)^46 + 178
39996318603553048412869885952*(sqrt(sqrt(x) + 1) + 1)^45 - 221074572906023619230346738925568*(sqrt(sqrt(x) + 1
) + 1)^44 + 153320643700628673330625866891264*(sqrt(sqrt(x) + 1) + 1)^43 - 26652891419311593866038343630848*(s
qrt(sqrt(x) + 1) + 1)^42 - 34964525177019636722858108911616*(sqrt(sqrt(x) + 1) + 1)^41 + 316821139447942895188
35974275072*(sqrt(sqrt(x) + 1) + 1)^40 - 9233374080713604069270669492224*(sqrt(sqrt(x) + 1) + 1)^39 - 38335385
80458548431139339501568*(sqrt(sqrt(x) + 1) + 1)^38 + 5085184419428714337736452997120*(sqrt(sqrt(x) + 1) + 1)^3
7 - 1982823679057600833030660816896*(sqrt(sqrt(x) + 1) + 1)^36 - 262480534359793423136287883264*(sqrt(sqrt(x)
+ 1) + 1)^35 + 711320861924448823343914680320*(sqrt(sqrt(x) + 1) + 1)^34 - 328697875402249596865599242240*(sqr
t(sqrt(x) + 1) + 1)^33 - 16461430004162620889537183744*(sqrt(sqrt(x) + 1) + 1)^32 + 95428601176521400977360683
008*(sqrt(sqrt(x) + 1) + 1)^31 - 41349359848761873167586164736*(sqrt(sqrt(x) + 1) + 1)^30 - 486397745655754356
1269084160*(sqrt(sqrt(x) + 1) + 1)^29 + 11544647980057943904629669888*(sqrt(sqrt(x) + 1) + 1)^28 - 33335243429
70261558455762944*(sqrt(sqrt(x) + 1) + 1)^27 - 1191572683417725493401812992*(sqrt(sqrt(x) + 1) + 1)^26 + 10262
32209353398029110476800*(sqrt(sqrt(x) + 1) + 1)^25 - 93481383755679278573023232*(sqrt(sqrt(x) + 1) + 1)^24 - 1
46062154264152631122427904*(sqrt(sqrt(x) + 1) + 1)^23 + 51788232298428869952700416*(sqrt(sqrt(x) + 1) + 1)^22
+ 9053298579516313975259136*(sqrt(sqrt(x) + 1) + 1)^21 - 8934958448492427163846656*(sqrt(sqrt(x) + 1) + 1)^20
+ 641389659530470477504512*(sqrt(sqrt(x) + 1) + 1)^19 + 915449581849135293882368*(sqrt(sqrt(x) + 1) + 1)^18 -
220733028492743248314368*(sqrt(sqrt(x) + 1) + 1...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \,d x \end {gather*}

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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