Optimal. Leaf size=160 \[ \frac {16}{5} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{5/2}-\frac {24}{7} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{7/2}+8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{9/2}-\frac {160}{11} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{11/2}+\frac {144}{13} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{13/2}-\frac {56}{15} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{15/2}+\frac {8}{17} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{17/2} \]
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Rubi [A]
time = 0.20, 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 = {1632, 1634}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1632
Rule 1634
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx &=2 \text {Subst}\left (\int x \left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {-1+x}\right )\\ &=4 \text {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 \text {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 \text {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 = 0.11, size = 94, normalized size = 0.59 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 107, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {16 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {17}{2}}}{17}\) | \(107\) |
default | \(\frac {16 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{\frac {17}{2}}}{17}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 106, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 62, normalized size = 0.39 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 859 vs.
\(2 (106) = 212\).
time = 5.69, size = 859, normalized size = 5.37 \begin {gather*} \frac {8}{765765} \, {\left (7 \, {\left (6435 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {17}{2}} - 58344 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {15}{2}} + 235620 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} - 556920 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} + 850850 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - 875160 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + 612612 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} - 291720 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} + 109395 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) + 119 \, {\left (429 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {15}{2}} - 3465 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} + 12285 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} - 25025 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} + 32175 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} - 27027 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} + 15015 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} - 6435 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) - 765 \, {\left (231 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} - 1638 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} + 5005 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - 8580 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + 9009 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} - 6006 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} + 3003 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) - 3315 \, {\left (63 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} - 385 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} + 990 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} - 1386 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} + 1155 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) + 9724 \, {\left (35 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - 180 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + 378 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} - 420 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) + 87516 \, {\left (5 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} - 21 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} + 35 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) - 102102 \, {\left (3 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} - 10 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right ) - 510510 \, {\left ({\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1}\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x - 1} + 1\right )}^{2} - 8 \, \sqrt {x - 1} - 7\right )\right )} \mathrm {sgn}\left (4 \, x - 7\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\sqrt {\sqrt {\sqrt {x-1}+1}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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