Optimal. Leaf size=83 \[ \frac {1}{96} \sqrt {x+\sqrt {x+1}} (8 x+27)+\frac {1}{48} \sqrt {x+1} (24 x-41) \sqrt {x+\sqrt {x+1}}-\frac {35}{64} \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]
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Rubi [A] time = 0.21, antiderivative size = 103, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1661, 640, 612, 621, 206} \begin {gather*} \frac {1}{2} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}-\frac {5}{12} \left (x+\sqrt {x+1}\right )^{3/2}-\frac {7}{32} \left (2 \sqrt {x+1}+1\right ) \sqrt {x+\sqrt {x+1}}+\frac {35}{64} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \frac {x \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \left (-1+x^2\right ) \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{2} \operatorname {Subst}\left (\int \left (-3-\frac {5 x}{2}\right ) \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {5}{12} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{2} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {7}{8} \operatorname {Subst}\left (\int \sqrt {-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {5}{12} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{2} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {7}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {35}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {5}{12} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{2} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {7}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {35}{32} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {5}{12} \left (x+\sqrt {1+x}\right )^{3/2}+\frac {1}{2} \sqrt {1+x} \left (x+\sqrt {1+x}\right )^{3/2}-\frac {7}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )+\frac {35}{64} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 0.92 \begin {gather*} \frac {1}{96} \sqrt {x+\sqrt {x+1}} \left (8 x \left (6 \sqrt {x+1}+1\right )-82 \sqrt {x+1}+27\right )+\frac {35}{64} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 74, normalized size = 0.89 \begin {gather*} \frac {1}{96} \sqrt {x+\sqrt {1+x}} \left (19-130 \sqrt {1+x}+8 (1+x)+48 (1+x)^{3/2}\right )-\frac {35}{64} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 64, normalized size = 0.77 \begin {gather*} \frac {1}{96} \, {\left (2 \, {\left (24 \, x - 41\right )} \sqrt {x + 1} + 8 \, x + 27\right )} \sqrt {x + \sqrt {x + 1}} + \frac {35}{128} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 62, normalized size = 0.75 \begin {gather*} \frac {1}{96} \, {\left (2 \, {\left (4 \, \sqrt {x + 1} {\left (6 \, \sqrt {x + 1} + 1\right )} - 65\right )} \sqrt {x + 1} + 19\right )} \sqrt {x + \sqrt {x + 1}} - \frac {35}{64} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 68, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{12}-\frac {7 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{32}+\frac {35 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{64}\) | \(68\) |
default | \(\frac {\sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{12}-\frac {7 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{32}+\frac {35 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{64}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + \sqrt {x + 1}} x}{\sqrt {x + 1}}\,{d x} \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+\sqrt {x+1}}}{\sqrt {x+1}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x \sqrt {x + \sqrt {x + 1}}}{\sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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