Optimal. Leaf size=102 \[ \frac{24 b \text{PolyLog}\left (2,\frac{a x}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a}+\frac{48 b \text{PolyLog}\left (3,\frac{a x}{a x+b}\right )}{a}+x \log ^3\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{6 b \log \left (1-\frac{a x}{a x+b}\right ) \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )}{a} \]
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Rubi [A] time = 0.131848, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2486, 2488, 2506, 6610} \[ \frac{24 b \text{PolyLog}\left (2,1-\frac{b}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a}+\frac{48 b \text{PolyLog}\left (3,1-\frac{b}{a x+b}\right )}{a}+x \log ^3\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{6 b \log \left (\frac{b}{a x+b}\right ) \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2486
Rule 2488
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int \log ^3\left (\frac{c (b+a x)^2}{x^2}\right ) \, dx &=x \log ^3\left (\frac{c (b+a x)^2}{x^2}\right )+(6 b) \int \frac{\log ^2\left (\frac{c (b+a x)^2}{x^2}\right )}{b+a x} \, dx\\ &=-\frac{6 b \log \left (\frac{b}{b+a x}\right ) \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^3\left (\frac{c (b+a x)^2}{x^2}\right )-\frac{\left (24 b^2\right ) \int \frac{\log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{x (b+a x)} \, dx}{a}\\ &=-\frac{6 b \log \left (\frac{b}{b+a x}\right ) \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^3\left (\frac{c (b+a x)^2}{x^2}\right )+\frac{24 b \log \left (\frac{c (b+a x)^2}{x^2}\right ) \text{Li}_2\left (1-\frac{b}{b+a x}\right )}{a}+\frac{\left (48 b^2\right ) \int \frac{\text{Li}_2\left (1-\frac{b}{b+a x}\right )}{x (b+a x)} \, dx}{a}\\ &=-\frac{6 b \log \left (\frac{b}{b+a x}\right ) \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^3\left (\frac{c (b+a x)^2}{x^2}\right )+\frac{24 b \log \left (\frac{c (b+a x)^2}{x^2}\right ) \text{Li}_2\left (1-\frac{b}{b+a x}\right )}{a}+\frac{48 b \text{Li}_3\left (1-\frac{b}{b+a x}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0245463, size = 98, normalized size = 0.96 \[ \frac{24 b \text{PolyLog}\left (2,\frac{a x}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a}+\frac{48 b \text{PolyLog}\left (3,\frac{a x}{a x+b}\right )}{a}+x \log ^3\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{6 b \log \left (\frac{b}{a x+b}\right ) \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.504, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ({\frac{c \left ( ax+b \right ) ^{2}}{{x}^{2}}} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \,{\left (2 \,{\left (a x + b\right )} \log \left (a x + b\right )^{3} + 3 \,{\left (a x \log \left (c\right ) - 2 \, a x \log \left (x\right )\right )} \log \left (a x + b\right )^{2}\right )}}{a} + \int \frac{a x \log \left (c\right )^{3} + b \log \left (c\right )^{3} - 8 \,{\left (a x + b\right )} \log \left (x\right )^{3} + 12 \,{\left (a x \log \left (c\right ) + b \log \left (c\right )\right )} \log \left (x\right )^{2} + 6 \,{\left ({\left (\log \left (c\right )^{2} - 4 \, \log \left (c\right )\right )} a x + b \log \left (c\right )^{2} + 4 \,{\left (a x + b\right )} \log \left (x\right )^{2} - 4 \,{\left (a x{\left (\log \left (c\right ) - 2\right )} + b \log \left (c\right )\right )} \log \left (x\right )\right )} \log \left (a x + b\right ) - 6 \,{\left (a x \log \left (c\right )^{2} + b \log \left (c\right )^{2}\right )} \log \left (x\right )}{a x + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left (\frac{a^{2} c x^{2} + 2 \, a b c x + b^{2} c}{x^{2}}\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 6 b \int \frac{\log{\left (a^{2} c + \frac{2 a b c}{x} + \frac{b^{2} c}{x^{2}} \right )}^{2}}{a x + b}\, dx + x \log{\left (\frac{c \left (a x + b\right )^{2}}{x^{2}} \right )}^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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