Optimal. Leaf size=32 \[ -2 \sqrt {x}+x+\frac {4 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1371, 715, 632,
210} \begin {gather*} \frac {4 \text {ArcTan}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+x-2 \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 715
Rule 1371
Rubi steps
\begin {align*} \int \frac {x}{1+\sqrt {x}+x} \, dx &=2 \text {Subst}\left (\int \frac {x^3}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-1+x+\frac {1}{1+x+x^2}\right ) \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}+x+2 \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}+x-4 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt {x}\right )\\ &=-2 \sqrt {x}+x+\frac {4 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} -2 \sqrt {x}+x+\frac {4 \tan ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 26, normalized size = 0.81
method | result | size |
derivativedivides | \(x +\frac {4 \arctan \left (\frac {\left (1+2 \sqrt {x}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}-2 \sqrt {x}\) | \(26\) |
default | \(x +\frac {4 \arctan \left (\frac {\left (1+2 \sqrt {x}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}-2 \sqrt {x}\) | \(26\) |
trager | \(-1+x -2 \sqrt {x}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \sqrt {x}+x -1}{\RootOf \left (\textit {\_Z}^{2}+3\right ) x -\RootOf \left (\textit {\_Z}^{2}+3\right )-3 x -3}\right )}{3}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 25, normalized size = 0.78 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + x - 2 \, \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 27, normalized size = 0.84 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x} + \frac {1}{3} \, \sqrt {3}\right ) + x - 2 \, \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 37, normalized size = 1.16 \begin {gather*} - 2 \sqrt {x} + x + \frac {4 \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 = 3.29, size = 25, normalized size = 0.78 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + x - 2 \, \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 27, normalized size = 0.84 \begin {gather*} x+\frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,\sqrt {x}}{3}\right )}{3}-2\,\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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