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 = 0.023297, antiderivative size = 32, normalized size of antiderivative = 1., 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 1357
Rule 701
Rule 618
Rule 204
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*}
Mathematica [A] time = 0.009687, size = 32, normalized size = 1. \[ x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.8 \begin{align*} x+{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,\sqrt{x} \right ) } \right ) }-2\,\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56694, size = 34, normalized size = 1.06 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + x - 2 \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4492, size = 96, normalized size = 3. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.228774, size = 37, normalized size = 1.16 \begin{align*} - 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07596, size = 34, normalized size = 1.06 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + x - 2 \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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