Optimal. Leaf size=29 \[ \frac{x \log (x)}{a (a+b x)}-\frac{\log (a+b x)}{a b} \]
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Rubi [A] time = 0.013455, antiderivative size = 29, 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 = {2314, 31} \[ \frac{x \log (x)}{a (a+b x)}-\frac{\log (a+b x)}{a b} \]
Antiderivative was successfully verified.
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Rule 2314
Rule 31
Rubi steps
\begin{align*} \int \frac{\log (x)}{(a+b x)^2} \, dx &=\frac{x \log (x)}{a (a+b x)}-\frac{\int \frac{1}{a+b x} \, dx}{a}\\ &=\frac{x \log (x)}{a (a+b x)}-\frac{\log (a+b x)}{a b}\\ \end{align*}
Mathematica [A] time = 0.0164849, size = 27, normalized size = 0.93 \[ \frac{\frac{x \log (x)}{a+b x}-\frac{\log (a+b x)}{b}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 30, normalized size = 1. \begin{align*}{\frac{x\ln \left ( x \right ) }{a \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) }{ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93178, size = 51, normalized size = 1.76 \begin{align*} -\frac{\frac{\log \left (b x + a\right )}{a} - \frac{\log \left (x\right )}{a}}{b} - \frac{\log \left (x\right )}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22414, size = 77, normalized size = 2.66 \begin{align*} \frac{b x \log \left (x\right ) -{\left (b x + a\right )} \log \left (b x + a\right )}{a b^{2} x + a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.358122, size = 24, normalized size = 0.83 \begin{align*} - \frac{\log{\left (x \right )}}{a b + b^{2} x} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09642, size = 49, normalized size = 1.69 \begin{align*} -\frac{\log \left (x\right )}{{\left (b x + a\right )} b} + \frac{\log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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