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 = 0.009634, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 2389
Rule 2295
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*}
Mathematica [A] time = 0.0044784, 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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Maple [A] time = 0.001, size = 32, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( bx+a \right ) x}{2}}+{\frac{\ln \left ( bx+a \right ) a}{2\,b}}-{\frac{x}{2}}-{\frac{a}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02115, size = 31, normalized size = 1.24 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74772, size = 53, normalized size = 2.12 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.318819, size = 29, normalized size = 1.16 \begin{align*} - 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3112, size = 31, normalized size = 1.24 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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