Optimal. Leaf size=190 \[ \frac {16}{17} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{17/2}-\frac {112}{15} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{15/2}+\frac {288}{13} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{13/2}-\frac {320}{11} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{11/2}+\frac {112}{9} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{9/2}+\frac {48}{7} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{7/2}-\frac {32}{5} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{5/2} \]
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Rubi [A] time = 0.37, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1618, 1620} \[ \frac {16}{17} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{17/2}-\frac {112}{15} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{15/2}+\frac {288}{13} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{13/2}-\frac {320}{11} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{11/2}+\frac {112}{9} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{9/2}+\frac {48}{7} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{7/2}-\frac {32}{5} \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 1618
Rule 1620
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {1+\sqrt {1+\sqrt {x}}}} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {x}\right )\\ &=4 \operatorname {Subst}\left (\int x \left (-1+x^2\right ) \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {x}}\right )\\ &=8 \operatorname {Subst}\left (\int x^3 \sqrt {1+x} \left (-2+x^2\right ) \left (-1+x^2\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {x}}}\right )\\ &=8 \operatorname {Subst}\left (\int x^3 (1+x)^{3/2} \left (2-2 x-x^2+x^3\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (-2 (1+x)^{3/2}+3 (1+x)^{5/2}+7 (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+\sqrt {x}}}\right )\\ &=-\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}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 135, normalized size = 0.71 \[ \frac {16 \left (\sqrt {\sqrt {\sqrt {x}+1}+1}+1\right )^{5/2} \left (231 \sqrt {x} \left (-377 \sqrt {\sqrt {\sqrt {x}+1}+1}+195 \sqrt {\sqrt {x}+1}+365\right )+8 \left (252 \sqrt {\sqrt {x}+1} \sqrt {\sqrt {\sqrt {x}+1}+1}+8642 \sqrt {\sqrt {\sqrt {x}+1}+1}-4865 \sqrt {\sqrt {x}+1}-8221\right )\right )}{765765} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 76, normalized size = 0.40 \[ \frac {16}{765765} \, {\left ({\left (231 \, \sqrt {x} - 1304\right )} \sqrt {\sqrt {x} + 1} + {\left ({\left (3003 \, \sqrt {x} - 4672\right )} \sqrt {\sqrt {x} + 1} - 3528 \, \sqrt {x} + 8752\right )} \sqrt {\sqrt {\sqrt {x} + 1} + 1} + 45045 \, x + 4613 \, \sqrt {x} - 28152\right )} \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.64 \[ -\frac {32 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {5}{2}}}{5}+\frac {48 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {7}{2}}}{7}+\frac {112 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {9}{2}}}{9}-\frac {320 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {11}{2}}}{11}+\frac {288 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {13}{2}}}{13}-\frac {112 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {15}{2}}}{15}+\frac {16 \left (1+\sqrt {1+\sqrt {\sqrt {x}+1}}\right )^{\frac {17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 120, normalized size = 0.63 \[ \frac {16}{17} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {17}{2}} - \frac {112}{15} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {15}{2}} + \frac {288}{13} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {13}{2}} - \frac {320}{11} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {11}{2}} + \frac {112}{9} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {9}{2}} + \frac {48}{7} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {32}{5} \, {\left (\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1\right )}^{\frac {5}{2}} \]
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 \[ \int \sqrt {\sqrt {\sqrt {\sqrt {x}+1}+1}+1} \,d x \]
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 \[ \int \sqrt {\sqrt {\sqrt {\sqrt {x} + 1} + 1} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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