Optimal. Leaf size=28 \[ x \log \left (\frac{c x^2}{(a x+b)^2}\right )-\frac{2 b \log (a x+b)}{a} \]
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Rubi [A] time = 0.0066862, 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} \[ x \log \left (\frac{c x^2}{(a x+b)^2}\right )-\frac{2 b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \log \left (\frac{c x^2}{(b+a x)^2}\right ) \, dx &=x \log \left (\frac{c x^2}{(b+a x)^2}\right )-(2 b) \int \frac{1}{b+a x} \, dx\\ &=x \log \left (\frac{c x^2}{(b+a x)^2}\right )-\frac{2 b \log (b+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0028988, size = 28, normalized size = 1. \[ x \log \left (\frac{c x^2}{(a x+b)^2}\right )-\frac{2 b \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.224, size = 79, normalized size = 2.8 \begin{align*} \ln \left ({\frac{c}{{a}^{2}} \left ({\frac{b}{ax+b}}-1 \right ) ^{2}} \right ) x+2\,{\frac{b\ln \left ( \left ( ax+b \right ) ^{-1} \right ) }{a}}-2\,{\frac{b}{a}\ln \left ({\frac{b}{ax+b}}-1 \right ) }+{\frac{b}{a}\ln \left ({\frac{c}{{a}^{2}} \left ({\frac{b}{ax+b}}-1 \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24105, size = 38, normalized size = 1.36 \begin{align*} x \log \left (\frac{c x^{2}}{{\left (a x + b\right )}^{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 = 1.96246, size = 88, normalized size = 3.14 \begin{align*} \frac{a x \log \left (\frac{c x^{2}}{a^{2} x^{2} + 2 \, a b x + b^{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 = 0.363537, size = 26, normalized size = 0.93 \begin{align*} x \log{\left (\frac{c x^{2}}{\left (a x + b\right )^{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 = 1.26297, size = 39, normalized size = 1.39 \begin{align*} x \log \left (\frac{c x^{2}}{{\left (a x + b\right )}^{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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