Optimal. Leaf size=83 \[ \frac {1}{96} (75+8 x) \sqrt {x+\sqrt {1+x}}+\frac {1}{48} \sqrt {1+x} (7+24 x) \sqrt {x+\sqrt {1+x}}+\frac {45}{64} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.24, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 654, 626,
635, 212} \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 {9}{32} \left (2 \sqrt {x+1}+1\right ) \sqrt {x+\sqrt {x+1}}-\frac {45}{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 212
Rule 626
Rule 635
Rule 654
Rule 756
Rubi steps
\begin {align*} \int \sqrt {1+x} \sqrt {x+\sqrt {1+x}} \, dx &=2 \text {Subst}\left (\int x^2 \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} \text {Subst}\left (\int \left (1-\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 {9}{8} \text {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 {9}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {45}{64} \text {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 {9}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {45}{32} \text {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 {9}{32} \sqrt {x+\sqrt {1+x}} \left (1+2 \sqrt {1+x}\right )-\frac {45}{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 = 0.09, size = 74, normalized size = 0.89 \begin {gather*} \frac {1}{96} \sqrt {x+\sqrt {1+x}} \left (67-34 \sqrt {1+x}+8 (1+x)+48 (1+x)^{3/2}\right )+\frac {45}{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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Maple [A]
time = 0.44, size = 68, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{12}+\frac {9 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{32}-\frac {45 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{64}\) | \(68\) |
default | \(\frac {\sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{2}-\frac {5 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{12}+\frac {9 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{32}-\frac {45 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{64}\) | \(68\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 64, normalized size = 0.77 \begin {gather*} \frac {1}{96} \, {\left (2 \, {\left (24 \, x + 7\right )} \sqrt {x + 1} + 8 \, x + 75\right )} \sqrt {x + \sqrt {x + 1}} + \frac {45}{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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x + 1} \sqrt {x + \sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 82, normalized size = 0.99 \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 {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + \frac {45}{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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \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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