Optimal. Leaf size=133 \[ \frac {8}{315} \sqrt {\sqrt {\sqrt {x+1}+1}+1} (35 x+27)+\sqrt {x+1} \left (\frac {8}{63} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}+\frac {8}{315} \sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )-\frac {64}{315} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \begin {gather*} \frac {8}{9} \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 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int x \sqrt {1+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \sqrt {1+\sqrt {x}} (-1+x) \, dx,x,1+\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int x \sqrt {1+x} \left (-1+x^2\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}+(1+x)^{7/2}\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}+\frac {8}{9} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.37 \begin {gather*} \frac {8}{315} \left (35 \sqrt {x+1}-65 \sqrt {\sqrt {x+1}+1}+61\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 85, normalized size = 0.64 \begin {gather*} \frac {8}{315} \sqrt {1+\sqrt {1+x}} \left (-8+5 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{315} \left (-8+\sqrt {1+x}+35 (1+x)\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 44, normalized size = 0.33 \begin {gather*} \frac {8}{315} \, {\left ({\left (5 \, \sqrt {x + 1} - 8\right )} \sqrt {\sqrt {x + 1} + 1} + 35 \, x + \sqrt {x + 1} + 27\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.76, size = 312, normalized size = 2.35 \begin {gather*} \frac {8}{315} \, {\left ({\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 ) + 9 \, {\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 ) - 21 \, {\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 ) - 105 \, {\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 + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 47, normalized size = 0.35
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}\) | \(47\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {9}{2}}}{9}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.31, size = 46, normalized size = 0.35 \begin {gather*} \frac {8}{9} \, {\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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \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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sympy [A] time = 2.27, size = 230, normalized size = 1.73 \begin {gather*} - \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{63 \pi } - \frac {\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } - \frac {\sqrt {2} \left (x + 1\right ) \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{9 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } + \frac {8 \sqrt {2} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{315 \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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