Optimal. Leaf size=29 \[ -x+\sqrt {x}+x \log \left (x+\sqrt {x}\right )-\log \left (\sqrt {x}+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2548, 376, 77} \[ -x+\sqrt {x}+x \log \left (x+\sqrt {x}\right )-\log \left (\sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 77
Rule 376
Rule 2548
Rubi steps
\begin {align*} \int \log \left (\sqrt {x}+x\right ) \, dx &=x \log \left (\sqrt {x}+x\right )-\int \frac {1+2 \sqrt {x}}{2+2 \sqrt {x}} \, dx\\ &=x \log \left (\sqrt {x}+x\right )-2 \operatorname {Subst}\left (\int \frac {x (1+2 x)}{2+2 x} \, dx,x,\sqrt {x}\right )\\ &=x \log \left (\sqrt {x}+x\right )-2 \operatorname {Subst}\left (\int \left (-\frac {1}{2}+x+\frac {1}{2 (1+x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}-x-\log \left (1+\sqrt {x}\right )+x \log \left (\sqrt {x}+x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ -x+\sqrt {x}+x \log \left (x+\sqrt {x}\right )-\log \left (\sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 31, normalized size = 1.07 \[ {\left (x + 1\right )} \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - 2 \, \log \left (\sqrt {x} + 1\right ) - \log \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 23, normalized size = 0.79 \[ x \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - \log \left (\sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 0.83 \[ x \ln \left (x +\sqrt {x}\right )-x -\ln \left (\sqrt {x}+1\right )+\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 23, normalized size = 0.79 \[ x \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - \log \left (\sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 23, normalized size = 0.79 \[ \sqrt {x}-\ln \left (\sqrt {x}+1\right )-x+x\,\ln \left (x+\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.63, size = 24, normalized size = 0.83 \[ \sqrt {x} + x \log {\left (\sqrt {x} + x \right )} - x - \log {\left (\sqrt {x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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