Optimal. Leaf size=32 \[ \frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-2 \sqrt {x} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-2 \sqrt {x} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \log (a+b x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \log (x) \, dx,x,a+b \sqrt {x}\right )}{b}\\ &=-2 \sqrt {x}+\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.03 \[ 2 \left (\frac {\left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 28, normalized size = 0.88 \[ \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 31, normalized size = 0.97 \[ \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.25 \[ 2 \sqrt {x}\, \ln \left (b \sqrt {x}+a \right )+\frac {2 a \ln \left (b \sqrt {x}+a \right )}{b}-2 \sqrt {x}-\frac {2 a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 31, normalized size = 0.97 \[ \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 33, normalized size = 1.03 \[ 2\,\sqrt {x}\,\ln \left (a+b\,\sqrt {x}\right )-2\,\sqrt {x}+\frac {2\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 133, normalized size = 4.16 \[ \begin {cases} \frac {2 a^{2} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 a^{2}}{a b + b^{2} \sqrt {x}} + \frac {4 a b \sqrt {x} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 b^{2} x \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} - \frac {2 b^{2} x}{a b + b^{2} \sqrt {x}} & \text {for}\: b \neq 0 \\2 \sqrt {x} \log {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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