Integrand size = 21, antiderivative size = 160 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {16}{5} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{5/2}-\frac {24}{7} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{7/2}+8 \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{9/2}-\frac {160}{11} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{11/2}+\frac {144}{13} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{13/2}-\frac {56}{15} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{15/2}+\frac {8}{17} \left (1+\sqrt {1+\sqrt {-1+x}}\right )^{17/2} \]
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Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1632, 1634} \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8}{17} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{17/2}-\frac {56}{15} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{15/2}+\frac {144}{13} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{13/2}-\frac {160}{11} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{11/2}+8 \left (\sqrt {\sqrt {x-1}+1}+1\right )^{9/2}-\frac {24}{7} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{7/2}+\frac {16}{5} \left (\sqrt {\sqrt {x-1}+1}+1\right )^{5/2} \]
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Rule 1632
Rule 1634
Rubi steps \begin{align*} \text {integral}& = 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*}
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8 \sqrt {1+\sqrt {1+\sqrt {-1+x}}} \left (-8872+1109 \sqrt {-1+x}+28231 (-1+x)+77 (-1+x)^{3/2}+15015 (-1+x)^2+\sqrt {1+\sqrt {-1+x}} \left (-7696+4544 \sqrt {-1+x}+7 \left (-168+143 \sqrt {-1+x}\right ) x\right )\right )}{255255} \]
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Time = 0.44 (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 | \(\frac {16 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {5}{2}}}{5}-\frac {24 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {7}{2}}}{7}+8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {9}{2}}-\frac {160 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {11}{2}}}{11}+\frac {144 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {13}{2}}}{13}-\frac {56 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {15}{2}}}{15}+\frac {8 \left (1+\sqrt {1+\sqrt {x -1}}\right )^{\frac {17}{2}}}{17}\) | \(107\) |
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.39 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8}{255255} \, {\left (15015 \, x^{2} + {\left (77 \, x + 1032\right )} \sqrt {x - 1} + {\left ({\left (1001 \, x + 4544\right )} \sqrt {x - 1} - 1176 \, x - 7696\right )} \sqrt {\sqrt {x - 1} + 1} - 1799 \, x - 22088\right )} \sqrt {\sqrt {\sqrt {x - 1} + 1} + 1} \]
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Time = 0.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {17}{2}}}{17} - \frac {56 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {15}{2}}}{15} + \frac {144 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {13}{2}}}{13} - \frac {160 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {11}{2}}}{11} + 8 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {9}{2}} - \frac {24 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {7}{2}}}{7} + \frac {16 \left (\sqrt {\sqrt {x - 1} + 1} + 1\right )^{\frac {5}{2}}}{5} \]
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Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\frac {8}{17} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {56}{15} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {144}{13} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {160}{11} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {11}{2}} + 8 \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {9}{2}} - \frac {24}{7} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {7}{2}} + \frac {16}{5} \, {\left (\sqrt {\sqrt {x - 1} + 1} + 1\right )}^{\frac {5}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (106) = 212\).
Time = 0.43 (sec) , antiderivative size = 859, normalized size of antiderivative = 5.37 \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\text {Too large to display} \]
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Timed out. \[ \int \sqrt {1+\sqrt {1+\sqrt {-1+x}}} x \, dx=\int x\,\sqrt {\sqrt {\sqrt {x-1}+1}+1} \,d x \]
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