Optimal. Leaf size=32 \[ x-2 \sqrt {x}+\frac {4 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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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 = {1357, 701, 618, 204} \[ x-2 \sqrt {x}+\frac {4 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 701
Rule 1357
Rubi steps
\begin {align*} \int \frac {x}{1+\sqrt {x}+x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-1+x+\frac {1}{1+x+x^2}\right ) \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}+x+2 \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}+x-4 \operatorname {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 = 0.01, size = 32, normalized size = 1.00 \[ x-2 \sqrt {x}+\frac {4 \tan ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 27, normalized size = 0.84 \[ \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x} + \frac {1}{3} \, \sqrt {3}\right ) + x - 2 \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 25, normalized size = 0.78 \[ \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + x - 2 \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.81 \[ x +\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 \sqrt {x}+1\right ) \sqrt {3}}{3}\right )}{3}-2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.13, size = 25, normalized size = 0.78 \[ \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + 1\right )}\right ) + x - 2 \, \sqrt {x} \]
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 \[ x+\frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,\sqrt {x}}{3}\right )}{3}-2\,\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 37, normalized size = 1.16 \[ - 2 \sqrt {x} + x + \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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