Optimal. Leaf size=25 \[ \frac {(a+b x) \log \left (\sqrt {a+b x}\right )}{b}-\frac {x}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2389, 2295} \[ \frac {(a+b x) \log \left (\sqrt {a+b x}\right )}{b}-\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rubi steps
\begin {align*} \int \log \left (\sqrt {a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \log \left (\sqrt {x}\right ) \, dx,x,a+b x\right )}{b}\\ &=-\frac {x}{2}+\frac {(a+b x) \log \left (\sqrt {a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 0.92 \[ \frac {1}{2} \left (\frac {(a+b x) \log (a+b x)}{b}-x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 22, normalized size = 0.88 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 23, normalized size = 0.92 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 32, normalized size = 1.28 \[ \frac {x \ln \left (b x +a \right )}{2}+\frac {a \ln \left (b x +a \right )}{2 b}-\frac {x}{2}-\frac {a}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 23, normalized size = 0.92 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 25, normalized size = 1.00 \[ \frac {x\,\ln \left (a+b\,x\right )}{2}-\frac {x}{2}+\frac {a\,\ln \left (a+b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 29, normalized size = 1.16 \[ - b \left (- \frac {a \log {\left (a + b x \right )}}{2 b^{2}} + \frac {x}{2 b}\right ) + \frac {x \log {\left (a + b x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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