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

Optimal. Leaf size=160 \[ \frac {8}{17} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{17/2}-\frac {56}{15} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{15/2}+\frac {144}{13} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{13/2}-\frac {160}{11} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{11/2}+8 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{9/2}-\frac {24}{7} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{7/2}+\frac {16}{5} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2} \]

[Out]

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)

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Rubi [A]  time = 0.28, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1618, 1620} \[ \frac {8}{17} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{17/2}-\frac {56}{15} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{15/2}+\frac {144}{13} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{13/2}-\frac {160}{11} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{11/2}+8 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{9/2}-\frac {24}{7} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{7/2}+\frac {16}{5} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]

[Out]

(16*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2))/5 - (24*(1 + Sqrt[1 + Sqrt[-1 + x]])^(7/2))/7 + 8*(1 + Sqrt[1 + Sqrt[-
1 + x]])^(9/2) - (160*(1 + Sqrt[1 + Sqrt[-1 + x]])^(11/2))/11 + (144*(1 + Sqrt[1 + Sqrt[-1 + x]])^(13/2))/13 -
(56*(1 + Sqrt[1 + Sqrt[-1 + x]])^(15/2))/15 + (8*(1 + Sqrt[1 + Sqrt[-1 + 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+x}}} x \, dx &=2 \operatorname {Subst}\left (\int x \left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {-1+x}\right )\\ &=4 \operatorname {Subst}\left (\int x \sqrt {1+x} \left (-1+x^2\right ) \left (1+\left (-1+x^2\right )^2\right ) \, dx,x,\sqrt {1+\sqrt {-1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int x (1+x)^{3/2} \left (-2+2 x+2 x^2-2 x^3-x^4+x^5\right ) \, dx,x,\sqrt {1+\sqrt {-1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (2 (1+x)^{3/2}-3 (1+x)^{5/2}+9 (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+x}}\right )\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 103, normalized size = 0.64 \[ \frac {8 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2} \left (8 \left (84 \sqrt {x-1} \sqrt {\sqrt {x-1}+1}-3030 \sqrt {\sqrt {x-1}+1}+1715 \sqrt {x-1}+2591\right )+77 \left (-377 \sqrt {\sqrt {x-1}+1}+195 \sqrt {x-1}+365\right ) x\right )}{255255} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2)*(8*(2591 - 3030*Sqrt[1 + Sqrt[-1 + x]] + 1715*Sqrt[-1 + x] + 84*Sqrt[1 +
Sqrt[-1 + x]]*Sqrt[-1 + x]) + 77*(365 - 377*Sqrt[1 + Sqrt[-1 + x]] + 195*Sqrt[-1 + x])*x))/255255

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fricas [A]  time = 0.47, size = 62, normalized size = 0.39 \[ \frac {8}{255255} \, {\left (15015 \, x^{2} + {\left (77 \, x + 1032\right )} \sqrt {x - 1} + {\left ({\left (1001 \, x + 4544\right )} \sqrt {x - 1} - 1176 \, x - 7696\right )} \sqrt {\sqrt {x - 1} + 1} - 1799 \, x - 22088\right )} \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+(1+(-1+x)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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(sqrt(x - 1) + 1) + 1)

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giac [B]  time = 15.54, size = 859, normalized size = 5.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+(1+(-1+x)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

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(sq
rt(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(sqr
t(x - 1) + 1) + 1)^(11/2) - 25025*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) + 32175*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2)
- 27027*(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) - 1
638*(sqrt(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*(sqrt(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*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) - 180*(sqrt(sqrt(x - 1) + 1) +
1)^(7/2) + 378*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2) - 420*(sqrt(sqrt(x - 1) + 1) + 1)^(3/2) + 315*sqrt(sqrt(sqrt(
x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) - 7) + 87516*(5*(sqrt(sqrt(x - 1) + 1) + 1)^(7/2)
- 21*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2) + 35*(sqrt(sqrt(x - 1) + 1) + 1)^(3/2) - 35*sqrt(sqrt(sqrt(x - 1) + 1)
+ 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) - 7) - 102102*(3*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2) - 10*(sqrt(
sqrt(x - 1) + 1) + 1)^(3/2) + 15*sqrt(sqrt(sqrt(x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2 - 8*sqrt(x - 1) -
7) - 510510*((sqrt(sqrt(x - 1) + 1) + 1)^(3/2) - 3*sqrt(sqrt(sqrt(x - 1) + 1) + 1))*sgn(4*(sqrt(x - 1) + 1)^2
- 8*sqrt(x - 1) - 7))*sgn(4*x - 7)

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maple [A]  time = 0.01, size = 107, normalized size = 0.67 \[ \frac {16 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)-16
0/11*(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))^
(15/2)+8/17*(1+(1+(x-1)^(1/2))^(1/2))^(17/2)

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maxima [A]  time = 0.89, size = 106, normalized size = 0.66 \[ \frac {8}{17} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {56}{15} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {144}{13} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {160}{11} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} + 8 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - \frac {24}{7} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + \frac {16}{5} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+(1+(-1+x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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*(sqr
t(sqrt(x - 1) + 1) + 1)^(7/2) + 16/5*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(((x - 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 \[ \int x \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+(1+(-1+x)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(x*sqrt(sqrt(sqrt(x - 1) + 1) + 1), x)

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