Optimal. Leaf size=28 \[ x \log \left (\frac{c (a x+b)^2}{x^2}\right )+\frac{2 b \log (a x+b)}{a} \]
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Rubi [A] time = 0.0068001, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2486, 31} \[ x \log \left (\frac{c (a x+b)^2}{x^2}\right )+\frac{2 b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \log \left (\frac{c (b+a x)^2}{x^2}\right ) \, dx &=x \log \left (\frac{c (b+a x)^2}{x^2}\right )+(2 b) \int \frac{1}{b+a x} \, dx\\ &=\frac{2 b \log (b+a x)}{a}+x \log \left (\frac{c (b+a x)^2}{x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0025771, size = 28, normalized size = 1. \[ x \log \left (\frac{c (a x+b)^2}{x^2}\right )+\frac{2 b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 40, normalized size = 1.4 \begin{align*} x\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{2} \right ) -2\,{\frac{b\ln \left ({x}^{-1} \right ) }{a}}+2\,{\frac{b}{a}\ln \left ( a+{\frac{b}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19178, size = 38, normalized size = 1.36 \begin{align*} x \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right ) + \frac{2 \, b \log \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 = 1.87655, size = 93, normalized size = 3.32 \begin{align*} \frac{a x \log \left (\frac{a^{2} c x^{2} + 2 \, a b c x + b^{2} c}{x^{2}}\right ) + 2 \, b \log \left (a x + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.311584, size = 26, normalized size = 0.93 \begin{align*} x \log{\left (\frac{c \left (a x + b\right )^{2}}{x^{2}} \right )} + \frac{2 b \log{\left (a x + b \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14719, size = 39, normalized size = 1.39 \begin{align*} x \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right ) + \frac{2 \, b \log \left ({\left | a x + b \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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