Optimal. Leaf size=42 \[ 2 \sqrt {x}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}}-\log \left (1+\sqrt {x}+x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1371, 717, 648,
632, 210, 642} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+2 \sqrt {x}-\log \left (x+\sqrt {x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rule 1371
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{1+\sqrt {x}+x} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}+2 \text {Subst}\left (\int \frac {-1-x}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}-\text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt {x}\right )-\text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}-\log \left (1+\sqrt {x}+x\right )+2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt {x}\right )\\ &=2 \sqrt {x}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}}-\log \left (1+\sqrt {x}+x\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} 2 \sqrt {x}-\frac {2 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}}-\log \left (1+\sqrt {x}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 34, normalized size = 0.81
method | result | size |
derivativedivides | \(-\ln \left (1+x +\sqrt {x}\right )-\frac {2 \arctan \left (\frac {\left (1+2 \sqrt {x}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+2 \sqrt {x}\) | \(34\) |
default | \(-\ln \left (1+x +\sqrt {x}\right )-\frac {2 \arctan \left (\frac {\left (1+2 \sqrt {x}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+2 \sqrt {x}\) | \(34\) |
trager | \(2 \sqrt {x}+2 \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (1+x +\sqrt {x}\right )-2 \ln \left (3 \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 \sqrt {x}+x -2\right ) \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-\ln \left (3 \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \RootOf \left (3 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 \sqrt {x}+x -2\right )\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 33, normalized size = 0.79 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 35, normalized size = 0.83 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x} + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 49, normalized size = 1.17 \begin {gather*} 2 \sqrt {x} - \log {\left (4 \sqrt {x} + 4 x + 4 \right )} - \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.68, size = 33, normalized size = 0.79 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.45, size = 35, normalized size = 0.83 \begin {gather*} 2\,\sqrt {x}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,\sqrt {x}}{3}\right )}{3}-\ln \left (x+\sqrt {x}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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