Integrand size = 21, antiderivative size = 160 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {16}{5} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{5/2}-\frac {24}{7} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{7/2}+8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{9/2}-\frac {160}{11} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{11/2}+\frac {144}{13} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{13/2}-\frac {56}{15} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{15/2}+\frac {8}{17} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{17/2} \] Output:
16/5*(1+(1+(-1+x)^(1/2))^(1/2))^(5/2)-24/7*(1+(1+(-1+x)^(1/2))^(1/2))^(7/2 )+8*(1+(1+(-1+x)^(1/2))^(1/2))^(9/2)-160/11*(1+(1+(-1+x)^(1/2))^(1/2))^(11 /2)+144/13*(1+(1+(-1+x)^(1/2))^(1/2))^(13/2)-56/15*(1+(1+(-1+x)^(1/2))^(1/ 2))^(15/2)+8/17*(1+(1+(-1+x)^(1/2))^(1/2))^(17/2)
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8 \sqrt {1+\sqrt {1+\sqrt {-1+x}}} \left (-8872+1109 \sqrt {-1+x}+28231 (-1+x)+77 (-1+x)^{3/2}+15015 (-1+x)^2+\sqrt {1+\sqrt {-1+x}} \left (-7696+4544 \sqrt {-1+x}+7 \left (-168+143 \sqrt {-1+x}\right ) x\right )\right )}{255255} \] Input:
Integrate[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]
Output:
(8*Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*(-8872 + 1109*Sqrt[-1 + x] + 28231*(-1 + x) + 77*(-1 + x)^(3/2) + 15015*(-1 + x)^2 + Sqrt[1 + Sqrt[-1 + x]]*(-76 96 + 4544*Sqrt[-1 + x] + 7*(-168 + 143*Sqrt[-1 + x])*x)))/255255
Time = 1.03 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7267, 7267, 25, 2003, 2091, 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.
\(\displaystyle \int \sqrt {\sqrt {\sqrt {x-1}+1}+1} x \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \sqrt {\sqrt {\sqrt {x-1}+1}+1} \sqrt {x-1} xd\sqrt {x-1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 4 \int -\sqrt {\sqrt {\sqrt {x-1}+1}+1} \left ((x-2)^2+1\right ) \sqrt {\sqrt {x-1}+1} (2-x)d\sqrt {\sqrt {x-1}+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \sqrt {\sqrt {\sqrt {x-1}+1}+1} \left ((x-2)^2+1\right ) \sqrt {\sqrt {x-1}+1} (2-x)d\sqrt {\sqrt {x-1}+1}\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle -4 \int \left (1-\sqrt {\sqrt {x-1}+1}\right ) \left (\sqrt {\sqrt {x-1}+1}+1\right )^{3/2} \left ((x-2)^2+1\right ) \sqrt {\sqrt {x-1}+1}d\sqrt {\sqrt {x-1}+1}\) |
\(\Big \downarrow \) 2091 |
\(\displaystyle -4 \int \left (1-\sqrt {\sqrt {x-1}+1}\right ) \left (\sqrt {\sqrt {x-1}+1}+1\right )^{3/2} \sqrt {\sqrt {x-1}+1} \left ((x-1)^2-2 (x-1)+2\right )d\sqrt {\sqrt {x-1}+1}\) |
\(\Big \downarrow \) 2115 |
\(\displaystyle -4 \int \left (-\left (\sqrt {\sqrt {x-1}+1}+1\right )^{15/2}+7 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{13/2}-18 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{11/2}+20 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{9/2}-9 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{7/2}+3 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2}-2 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{3/2}\right )d\sqrt {\sqrt {x-1}+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (\frac {2}{17} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{17/2}-\frac {14}{15} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{15/2}+\frac {36}{13} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{13/2}-\frac {40}{11} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{11/2}+2 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{9/2}-\frac {6}{7} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{7/2}+\frac {4}{5} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2}\right )\) |
Input:
Int[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]
Output:
4*((4*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2))/5 - (6*(1 + Sqrt[1 + Sqrt[-1 + x ]])^(7/2))/7 + 2*(1 + Sqrt[1 + Sqrt[-1 + x]])^(9/2) - (40*(1 + Sqrt[1 + Sq rt[-1 + x]])^(11/2))/11 + (36*(1 + Sqrt[1 + Sqrt[-1 + x]])^(13/2))/13 - (1 4*(1 + Sqrt[1 + Sqrt[-1 + x]])^(15/2))/15 + (2*(1 + Sqrt[1 + Sqrt[-1 + 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_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x ] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
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 = 0.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {16 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {17}{2}}}{17}\) | \(107\) |
default | \(\frac {16 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {17}{2}}}{17}\) | \(107\) |
Input:
int((1+(1+(x-1)^(1/2))^(1/2))^(1/2)*x,x,method=_RETURNVERBOSE)
Output:
16/5*(1+(1+(x-1)^(1/2))^(1/2))^(5/2)-24/7*(1+(1+(x-1)^(1/2))^(1/2))^(7/2)+ 8*(1+(1+(x-1)^(1/2))^(1/2))^(9/2)-160/11*(1+(1+(x-1)^(1/2))^(1/2))^(11/2)+ 144/13*(1+(1+(x-1)^(1/2))^(1/2))^(13/2)-56/15*(1+(1+(x-1)^(1/2))^(1/2))^(1 5/2)+8/17*(1+(1+(x-1)^(1/2))^(1/2))^(17/2)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.39 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8}{255255} \, {\left (15015 \, x^{2} + {\left (77 \, x + 1032\right )} \sqrt {x - 1} + {\left ({\left (1001 \, x + 4544\right )} \sqrt {x - 1} - 1176 \, x - 7696\right )} \sqrt {\sqrt {x - 1} + 1} - 1799 \, x - 22088\right )} \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1} \] Input:
integrate((1+(1+(x-1)^(1/2))^(1/2))^(1/2)*x,x, algorithm="fricas")
Output:
8/255255*(15015*x^2 + (77*x + 1032)*sqrt(x - 1) + ((1001*x + 4544)*sqrt(x - 1) - 1176*x - 7696)*sqrt(sqrt(x - 1) + 1) - 1799*x - 22088)*sqrt(sqrt(sq rt(x - 1) + 1) + 1)
Time = 0.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {17}{2}}}{17} - \frac {56 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {15}{2}}}{15} + \frac {144 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {13}{2}}}{13} - \frac {160 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {11}{2}}}{11} + 8 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {9}{2}} - \frac {24 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {7}{2}}}{7} + \frac {16 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {5}{2}}}{5} \] Input:
integrate((1+(1+(x-1)**(1/2))**(1/2))**(1/2)*x,x)
Output:
8*(sqrt(sqrt(x - 1) + 1) + 1)**(17/2)/17 - 56*(sqrt(sqrt(x - 1) + 1) + 1)* *(15/2)/15 + 144*(sqrt(sqrt(x - 1) + 1) + 1)**(13/2)/13 - 160*(sqrt(sqrt(x - 1) + 1) + 1)**(11/2)/11 + 8*(sqrt(sqrt(x - 1) + 1) + 1)**(9/2) - 24*(sq rt(sqrt(x - 1) + 1) + 1)**(7/2)/7 + 16*(sqrt(sqrt(x - 1) + 1) + 1)**(5/2)/ 5
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8}{17} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {56}{15} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {144}{13} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {160}{11} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} + 8 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - \frac {24}{7} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + \frac {16}{5} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} \] Input:
integrate((1+(1+(x-1)^(1/2))^(1/2))^(1/2)*x,x, algorithm="maxima")
Output:
8/17*(sqrt(sqrt(x - 1) + 1) + 1)^(17/2) - 56/15*(sqrt(sqrt(x - 1) + 1) + 1 )^(15/2) + 144/13*(sqrt(sqrt(x - 1) + 1) + 1)^(13/2) - 160/11*(sqrt(sqrt(x - 1) + 1) + 1)^(11/2) + 8*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) - 24/7*(sqrt( sqrt(x - 1) + 1) + 1)^(7/2) + 16/5*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (106) = 212\).
Time = 0.18 (sec) , antiderivative size = 859, normalized size of antiderivative = 5.37 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\text {Too large to display} \] Input:
integrate((1+(1+(x-1)^(1/2))^(1/2))^(1/2)*x,x, algorithm="giac")
Output:
8/765765*(7*(6435*(sqrt(sqrt(x - 1) + 1) + 1)^(17/2) - 58344*(sqrt(sqrt(x - 1) + 1) + 1)^(15/2) + 235620*(sqrt(sqrt(x - 1) + 1) + 1)^(13/2) - 556920 *(sqrt(sqrt(x - 1) + 1) + 1)^(11/2) + 850850*(sqrt(sqrt(x - 1) + 1) + 1)^( 9/2) - 875160*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2) + 612612*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2) - 291720*(sqrt(sqrt(x - 1) + 1) + 1)^(3/2) + 109395*sqrt( sqrt(sqrt(x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) - 7) + 119*(429*(sqrt(sqrt(x - 1) + 1) + 1)^(15/2) - 3465*(sqrt(sqrt(x - 1) + 1) + 1)^(13/2) + 12285*(sqrt(sqrt(x - 1) + 1) + 1)^(11/2) - 25025*(sqrt(sq rt(x - 1) + 1) + 1)^(9/2) + 32175*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2) - 2702 7*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2) + 15015*(sqrt(sqrt(x - 1) + 1) + 1)^(3 /2) - 6435*sqrt(sqrt(sqrt(x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8* sqrt(x - 1) - 7) - 765*(231*(sqrt(sqrt(x - 1) + 1) + 1)^(13/2) - 1638*(sqr t(sqrt(x - 1) + 1) + 1)^(11/2) + 5005*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) - 8580*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2) + 9009*(sqrt(sqrt(x - 1) + 1) + 1)^ (5/2) - 6006*(sqrt(sqrt(x - 1) + 1) + 1)^(3/2) + 3003*sqrt(sqrt(sqrt(x - 1 ) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) - 7) - 3315*(63*(sq rt(sqrt(x - 1) + 1) + 1)^(11/2) - 385*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) + 990*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2) - 1386*(sqrt(sqrt(x - 1) + 1) + 1)^( 5/2) + 1155*(sqrt(sqrt(x - 1) + 1) + 1)^(3/2) - 693*sqrt(sqrt(sqrt(x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) - 7) + 9724*(35*(s...
Timed out. \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\int x\,\sqrt {\sqrt {\sqrt {x-1}+1}+1} \,d x \] Input:
int(x*(((x - 1)^(1/2) + 1)^(1/2) + 1)^(1/2),x)
Output:
int(x*(((x - 1)^(1/2) + 1)^(1/2) + 1)^(1/2), x)
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.51 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8 \sqrt {\sqrt {\sqrt {x -1}+1}+1}\, \left (1001 \sqrt {x -1}\, \sqrt {\sqrt {x -1}+1}\, x +4544 \sqrt {x -1}\, \sqrt {\sqrt {x -1}+1}-1176 \sqrt {\sqrt {x -1}+1}\, x -7696 \sqrt {\sqrt {x -1}+1}+77 \sqrt {x -1}\, x +1032 \sqrt {x -1}+15015 x^{2}-1799 x -22088\right )}{255255} \] Input:
int((1+(1+(x-1)^(1/2))^(1/2))^(1/2)*x,x)
Output:
(8*sqrt(sqrt(sqrt(x - 1) + 1) + 1)*(1001*sqrt(x - 1)*sqrt(sqrt(x - 1) + 1) *x + 4544*sqrt(x - 1)*sqrt(sqrt(x - 1) + 1) - 1176*sqrt(sqrt(x - 1) + 1)*x - 7696*sqrt(sqrt(x - 1) + 1) + 77*sqrt(x - 1)*x + 1032*sqrt(x - 1) + 1501 5*x**2 - 1799*x - 22088))/255255