Integrand size = 23, antiderivative size = 190 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=-\frac {32}{5} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{5/2}+\frac {48}{7} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{7/2}+\frac {112}{9} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{9/2}-\frac {320}{11} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{11/2}+\frac {288}{13} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{13/2}-\frac {112}{15} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{15/2}+\frac {16}{17} \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{17/2} \] Output:
-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)
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.88 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\frac {16 \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \left (-8 \left (3519-1094 \sqrt {1+\sqrt {1+\sqrt {x}}}+163 \sqrt {1+\sqrt {x}}+584 \sqrt {1+\sqrt {1+\sqrt {x}}} \sqrt {1+\sqrt {x}}\right )+7 \left (659-504 \sqrt {1+\sqrt {1+\sqrt {x}}}+33 \sqrt {1+\sqrt {x}}+429 \sqrt {1+\sqrt {1+\sqrt {x}}} \sqrt {1+\sqrt {x}}\right ) \sqrt {x}+45045 x\right )}{765765} \] Input:
Integrate[Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]],x]
Output:
(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]]]*Sqr t[1 + Sqrt[x]]) + 7*(659 - 504*Sqrt[1 + Sqrt[1 + Sqrt[x]]] + 33*Sqrt[1 + S qrt[x]] + 429*Sqrt[1 + Sqrt[1 + Sqrt[x]]]*Sqrt[1 + Sqrt[x]])*Sqrt[x] + 450 45*x))/765765
Time = 1.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {7267, 7267, 25, 7267, 2003, 2115, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \sqrt {x}d\sqrt {x}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 4 \int -\sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \sqrt {\sqrt {x}+1} (1-x)d\sqrt {\sqrt {x}+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \sqrt {\sqrt {x}+1} (1-x)d\sqrt {\sqrt {x}+1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 8 \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} (1-x) (2-x) x^{3/2}d\sqrt {\sqrt {\sqrt {x}+1}+1}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle 8 \int \left (1-\sqrt {\sqrt {\sqrt {x}+1}+1}\right ) \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{3/2} (2-x) x^{3/2}d\sqrt {\sqrt {\sqrt {x}+1}+1}\) |
| \(\Big \downarrow \) 2115 |
| \(\displaystyle 8 \int \left (\left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{15/2}-7 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{13/2}+18 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{11/2}-20 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{9/2}+7 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{7/2}+3 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{5/2}-2 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{3/2}\right )d\sqrt {\sqrt {\sqrt {x}+1}+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \left (\frac {2}{17} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{17/2}-\frac {14}{15} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{15/2}+\frac {36}{13} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{13/2}-\frac {40}{11} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{11/2}+\frac {14}{9} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{9/2}+\frac {6}{7} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{7/2}-\frac {4}{5} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{5/2}\right )\) |
Input:
Int[Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]]],x]
Output:
8*((-4*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(5/2))/5 + (6*(1 + Sqrt[1 + Sqrt[ 1 + Sqrt[x]]])^(7/2))/7 + (14*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(9/2))/9 - (40*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(11/2))/11 + (36*(1 + Sqrt[1 + Sqrt [1 + Sqrt[x]]])^(13/2))/13 - (14*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(15/2)) /15 + (2*(1 + Sqrt[1 + Sqrt[1 + Sqrt[x]]])^(17/2))/17)
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[m, n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.82 (sec) , antiderivative size = 121, normalized size of antiderivative = 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 | \(-\frac {32 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {5}{2}}}{5}+\frac {48 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {7}{2}}}{7}+\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {9}{2}}}{9}-\frac {320 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {11}{2}}}{11}+\frac {288 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {13}{2}}}{13}-\frac {112 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {15}{2}}}{15}+\frac {16 \left (1+\sqrt {1+\sqrt {1+\sqrt {x}}}\right )^{\frac {17}{2}}}{17}\) | \(121\) |
Input:
int((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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)
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.40 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\frac {16}{765765} \, {\left ({\left (231 \, \sqrt {x} - 1304\right )} \sqrt {\sqrt {x} + 1} + {\left ({\left (3003 \, \sqrt {x} - 4672\right )} \sqrt {\sqrt {x} + 1} - 3528 \, \sqrt {x} + 8752\right )} \sqrt {\sqrt {\sqrt {x} + 1} + 1} + 45045 \, x + 4613 \, \sqrt {x} - 28152\right )} \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1} \] Input:
integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="fricas")
Output:
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) + 45 045*x + 4613*sqrt(x) - 28152)*sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)
Time = 0.66 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.87 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\frac {16 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {17}{2}}}{17} - \frac {112 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {15}{2}}}{15} + \frac {288 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {13}{2}}}{13} - \frac {320 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {11}{2}}}{11} + \frac {112 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {9}{2}}}{9} + \frac {48 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {7}{2}}}{7} - \frac {32 \left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )^{\frac {5}{2}}}{5} \] Input:
integrate((1+(1+(1+x**(1/2))**(1/2))**(1/2))**(1/2),x)
Output:
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)/1 3 - 320*(sqrt(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
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.63 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\frac {16}{17} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {112}{15} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {288}{13} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {320}{11} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {11}{2}} + \frac {112}{9} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {9}{2}} + \frac {48}{7} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {32}{5} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {5}{2}} \] Input:
integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="maxima")
Output:
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*(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)
Leaf count of result is larger than twice the leaf count of optimal. 7916 vs. \(2 (120) = 240\).
Time = 39.83 (sec) , antiderivative size = 7916, normalized size of antiderivative = 41.66 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\text {Too large to display} \] Input:
integrate((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x, algorithm="giac")
Output:
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*(sqr t(sqrt(sqrt(x) + 1) + 1) + 1)^(9/2) - 875160*(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)^(7/2) + 612612*(sqrt(sqrt(sqrt(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 + 291326600895528960*(sqrt(sqrt(x) + 1) + 1)^90 - 8545580292935516160*(sqrt(sqrt(x) + 1) + 1)^89 + 183728762437276532736* (sqrt(sqrt(x) + 1) + 1)^88 - 3086556782054646743040*(sqrt(sqrt(x) + 1) + 1 )^87 + 42179809308639429132288*(sqrt(sqrt(x) + 1) + 1)^86 - 48197884682284 1400164352*(sqrt(sqrt(x) + 1) + 1)^85 + 4697911198078384159588352*(sqrt(sq rt(x) + 1) + 1)^84 - 39651330432185076620984320*(sqrt(sqrt(x) + 1) + 1)^83 + 293183639716003233721745408*(sqrt(sqrt(x) + 1) + 1)^82 - 19166563364402 69370174734336*(sqrt(sqrt(x) + 1) + 1)^81 + 11160164453620451334571425792* (sqrt(sqrt(x) + 1) + 1)^80 - 58223902019906429347317153792*(sqrt(sqrt(x) + 1) + 1)^79 + 273479024956137655533112918016*(sqrt(sqrt(x) + 1) + 1)^78 - 1160956607882993155309408616448*(sqrt(sqrt(x) + 1) + 1)^77 + 4467886822469 532994953426239488*(sqrt(sqrt(x) + 1) + 1)^76 - 15624039803063454614788052 615168*(sqrt(sqrt(x) + 1) + 1)^75 + 49728771914087708805425247813632*(s...
Timed out. \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \,d x \] Input:
int((((x^(1/2) + 1)^(1/2) + 1)^(1/2) + 1)^(1/2),x)
Output:
int((((x^(1/2) + 1)^(1/2) + 1)^(1/2) + 1)^(1/2), x)
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.48 \[ \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx=\frac {16 \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1}\, \left (3003 \sqrt {x}\, \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {\sqrt {x}+1}+1}-4672 \sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {\sqrt {x}+1}+1}-3528 \sqrt {x}\, \sqrt {\sqrt {\sqrt {x}+1}+1}+8752 \sqrt {\sqrt {\sqrt {x}+1}+1}+231 \sqrt {x}\, \sqrt {\sqrt {x}+1}-1304 \sqrt {\sqrt {x}+1}+4613 \sqrt {x}+45045 x -28152\right )}{765765} \] Input:
int((1+(1+(1+x^(1/2))^(1/2))^(1/2))^(1/2),x)
Output:
(16*sqrt(sqrt(sqrt(sqrt(x) + 1) + 1) + 1)*(3003*sqrt(x)*sqrt(sqrt(x) + 1)* sqrt(sqrt(sqrt(x) + 1) + 1) - 4672*sqrt(sqrt(x) + 1)*sqrt(sqrt(sqrt(x) + 1 ) + 1) - 3528*sqrt(x)*sqrt(sqrt(sqrt(x) + 1) + 1) + 8752*sqrt(sqrt(sqrt(x) + 1) + 1) + 231*sqrt(x)*sqrt(sqrt(x) + 1) - 1304*sqrt(sqrt(x) + 1) + 4613 *sqrt(x) + 45045*x - 28152))/765765