Optimal. Leaf size=82 \[ -\frac {\text {Li}_2\left (1-\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{a}+\frac {2 \text {Li}_3\left (1-\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{a}-\frac {2 \text {Li}_4\left (1-\frac {a}{a+b x}\right )}{a} \]
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Rubi [A] time = 0.17, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2506, 2508, 6610} \[ -\frac {\text {PolyLog}\left (2,1-\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{a}+\frac {2 \text {PolyLog}\left (3,1-\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{a}-\frac {2 \text {PolyLog}\left (4,1-\frac {a}{a+b x}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2506
Rule 2508
Rule 6610
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx &=-\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+2 \int \frac {\log \left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{x (a+b x)} \, dx\\ &=-\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \text {Li}_3\left (1-\frac {a}{a+b x}\right )}{a}-2 \int \frac {\text {Li}_3\left (1-\frac {a}{a+b x}\right )}{x (a+b x)} \, dx\\ &=-\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \text {Li}_3\left (1-\frac {a}{a+b x}\right )}{a}-\frac {2 \text {Li}_4\left (1-\frac {a}{a+b x}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 76, normalized size = 0.93 \[ -\frac {\text {Li}_2\left (\frac {b x}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{a}+\frac {2 \text {Li}_3\left (\frac {b x}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{a}-\frac {2 \text {Li}_4\left (\frac {b x}{a+b x}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {c x}{b x + a}\right )^{2} \log \left (\frac {a}{b x + a}\right )}{b x^{2} + a x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {c x}{b x + a}\right )^{2} \log \left (\frac {a}{b x + a}\right )}{{\left (b x + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\frac {a}{b x +a}\right ) \ln \left (\frac {c x}{b x +a}\right )^{2}}{\left (b x +a \right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\log \left (b x + a\right )^{4} - 4 \, \log \left (b x + a\right )^{3} \log \relax (x)}{4 \, a} + \int \frac {a \log \relax (a) \log \relax (c)^{2} + 2 \, a \log \relax (a) \log \relax (c) \log \relax (x) + a \log \relax (a) \log \relax (x)^{2} + {\left (a {\left (\log \relax (a) + 2 \, \log \relax (c)\right )} + {\left (3 \, b x + 2 \, a\right )} \log \relax (x)\right )} \log \left (b x + a\right )^{2} - {\left (2 \, a {\left (\log \relax (a) + \log \relax (c)\right )} \log \relax (x) + a \log \relax (x)^{2} + {\left (2 \, \log \relax (a) \log \relax (c) + \log \relax (c)^{2}\right )} a\right )} \log \left (b x + a\right )}{a b x^{2} + a^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (\frac {c\,x}{a+b\,x}\right )}^2\,\ln \left (\frac {a}{a+b\,x}\right )}{x\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\frac {a}{a + b x} \right )} \log {\left (\frac {c x}{a + b x} \right )}^{2}}{x \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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