Optimal. Leaf size=42 \[ 2 \sqrt {x}-\log \left (x+\sqrt {x}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, 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 = {1357, 703, 634, 618, 204, 628} \[ 2 \sqrt {x}-\log \left (x+\sqrt {x}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 703
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{1+\sqrt {x}+x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}+2 \operatorname {Subst}\left (\int \frac {-1-x}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}-\operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt {x}\right )-\operatorname {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 \operatorname {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 = 0.01, size = 42, normalized size = 1.00 \[ 2 \sqrt {x}-\log \left (x+\sqrt {x}+1\right )-\frac {2 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 35, normalized size = 0.83 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x} + \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 33, normalized size = 0.79 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.81 \[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \sqrt {x}+1\right ) \sqrt {3}}{3}\right )}{3}-\ln \left (x +\sqrt {x}+1\right )+2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 33, normalized size = 0.79 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + 2 \, \sqrt {x} - \log \left (x + \sqrt {x} + 1\right ) \]
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 \[ 2\,\sqrt {x}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,\sqrt {x}}{3}\right )}{3}-\ln \left (x+\sqrt {x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 49, normalized size = 1.17 \[ 2 \sqrt {x} - \log {\left (4 \sqrt {x} + 4 x + 4 \right )} - \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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