Optimal. Leaf size=47 \[ \frac {4}{15} \sqrt {\sqrt {x+1}+1} (3 x+1)+\frac {4}{15} \sqrt {x+1} \sqrt {\sqrt {x+1}+1} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 190, 43} \begin {gather*} \frac {4}{5} \left (\sqrt {x+1}+1\right )^{5/2}-\frac {4}{3} \left (\sqrt {x+1}+1\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \sqrt {1+\sqrt {1+x}} \, dx &=\operatorname {Subst}\left (\int \sqrt {1+\sqrt {x}} \, dx,x,1+x\right )\\ &=2 \operatorname {Subst}\left (\int x \sqrt {1+x} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.60 \begin {gather*} \frac {4}{15} \left (\sqrt {x+1}+1\right )^{3/2} \left (3 \sqrt {x+1}-2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 28, normalized size = 0.60 \begin {gather*} \frac {4}{15} \left (1+\sqrt {1+x}\right )^{3/2} \left (-2+3 \sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 0.45 \begin {gather*} \frac {4}{15} \, {\left (3 \, x + \sqrt {x + 1} + 1\right )} \sqrt {\sqrt {x + 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.23, size = 23, normalized size = 0.49 \begin {gather*} \frac {4}{5} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {5}{2}} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 17, normalized size = 0.36
method | result | size |
meijerg | \(\hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, 1\right ], \left [\frac {1}{2}, 2\right ], -x \right ) \sqrt {2}\, x\) | \(17\) |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}\) | \(24\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.32, size = 23, normalized size = 0.49 \begin {gather*} \frac {4}{5} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {5}{2}} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 16, normalized size = 0.34 \begin {gather*} \left (x+1\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ -\sqrt {x+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 184, normalized size = 3.91 \begin {gather*} \frac {12 \left (x + 1\right )^{\frac {7}{2}} \sqrt {\sqrt {x + 1} + 1}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} - \frac {4 \left (x + 1\right )^{\frac {5}{2}} \sqrt {\sqrt {x + 1} + 1}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} + \frac {8 \left (x + 1\right )^{\frac {5}{2}}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} + \frac {16 \left (x + 1\right )^{3} \sqrt {\sqrt {x + 1} + 1}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} - \frac {8 \left (x + 1\right )^{2} \sqrt {\sqrt {x + 1} + 1}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} + \frac {8 \left (x + 1\right )^{2}}{15 \left (x + 1\right )^{\frac {5}{2}} + 15 \left (x + 1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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