Optimal. Leaf size=119 \[ -\frac {4}{3} \sqrt {1+\sqrt {1+x}}+\sqrt {1+x} \left (\frac {2 (1+x)}{3}-\frac {4}{3} \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right ) \]
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Rubi [A]
time = 0.41, antiderivative size = 127, normalized size of antiderivative = 1.07, number of steps
used = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1642, 646, 31}
\begin {gather*} \frac {2}{3} \left (\sqrt {x+1}+1\right )^3-2 \left (\sqrt {x+1}+1\right )^2-\frac {4}{3} \left (\sqrt {x+1}+1\right )^{3/2}+2 \sqrt {x+1}+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1642
Rubi steps
\begin {align*} \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )}{-1+x^2+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {(-1+x) x^2 (1+x)^2 \left (-2+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \text {Subst}\left (\int \left (x-x^2-2 x^3+x^5+\frac {x}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+4 \text {Subst}\left (\int \frac {x}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 95, normalized size = 0.80 \begin {gather*} \frac {2}{15} \left (-10 \left (1+\sqrt {1+x}\right )^{3/2}+5 \left (1+(1+x)^{3/2}\right )-3 \left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+3 \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 85, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2+2 \sqrt {1+x}+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(85\) |
default | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2+2 \sqrt {1+x}+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 102, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 130, normalized size = 1.09 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} + \frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 66.54, size = 175, normalized size = 1.47 \begin {gather*} 2 \sqrt {x + 1} - \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} + \frac {2 \left (\sqrt {x + 1} + 1\right )^{3}}{3} - 2 \left (\sqrt {x + 1} + 1\right )^{2} - 8 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + 2 \log {\left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1} \right )} + 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 102, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 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 \frac {x\,\sqrt {x+1}}{x+\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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