Optimal. Leaf size=19 \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{1+\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-1+x+\frac {1}{1+x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}+x+2 \log \left (1+\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ x-2 \sqrt {x}+2 \log \left (\sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 15, normalized size = 0.79 \[ x - 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 15, normalized size = 0.79 \[ x - 2 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.84 \[ x +2 \ln \left (\sqrt {x}+1\right )-2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 22, normalized size = 1.16 \[ {\left (\sqrt {x} + 1\right )}^{2} - 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 1\right ) - 4 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 15, normalized size = 0.79 \[ x+2\,\ln \left (\sqrt {x}+1\right )-2\,\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 17, normalized size = 0.89 \[ - 2 \sqrt {x} + x + 2 \log {\left (\sqrt {x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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