Optimal. Leaf size=29 \[ \frac{x \log (x)}{b (a x+b)}-\frac{\log (a x+b)}{a b} \]
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Rubi [A] time = 0.0130004, 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} \[ \frac{x \log (x)}{b (a x+b)}-\frac{\log (a x+b)}{a b} \]
Antiderivative was successfully verified.
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Rule 2314
Rule 31
Rubi steps
\begin{align*} \int \frac{\log (x)}{(b+a x)^2} \, dx &=\frac{x \log (x)}{b (b+a x)}-\frac{\int \frac{1}{b+a x} \, dx}{b}\\ &=\frac{x \log (x)}{b (b+a x)}-\frac{\log (b+a x)}{a b}\\ \end{align*}
Mathematica [A] time = 0.0125981, size = 27, normalized size = 0.93 \[ \frac{\frac{x \log (x)}{a x+b}-\frac{\log (a x+b)}{a}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 30, normalized size = 1. \begin{align*}{\frac{x\ln \left ( x \right ) }{b \left ( ax+b \right ) }}-{\frac{\ln \left ( ax+b \right ) }{ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929854, size = 51, normalized size = 1.76 \begin{align*} -\frac{\frac{\log \left (a x + b\right )}{b} - \frac{\log \left (x\right )}{b}}{a} - \frac{\log \left (x\right )}{{\left (a x + b\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99178, size = 77, normalized size = 2.66 \begin{align*} \frac{a x \log \left (x\right ) -{\left (a x + b\right )} \log \left (a x + b\right )}{a^{2} b x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.362573, size = 24, normalized size = 0.83 \begin{align*} - \frac{\log{\left (x \right )}}{a^{2} x + a b} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10867, size = 49, normalized size = 1.69 \begin{align*} -\frac{\log \left (x\right )}{{\left (a x + b\right )} a} + \frac{\log \left ({\left | -\frac{b}{a x + b} + 1 \right |}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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