Optimal. Leaf size=519 \[ \frac {1}{12} \left (-24 \left (\text {Li}_4\left (-\frac {b x}{a}\right )+\text {Li}_4\left (\frac {b x}{a+b x}\right )-\text {Li}_4\left (\frac {b x}{a}+1\right )\right )+12 \text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log ^2\left (-\frac {b x}{a}\right )-2 \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log \left (-\frac {b x}{a}\right )+2 \log (x) \left (\log \left (\frac {b x}{a}+1\right )-\log (a+b x)\right )\right )-12 \text {Li}_2\left (\frac {b x}{a+b x}\right ) \log ^2\left (-\frac {b x}{a+b x}\right )+12 \text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {b x}{a+b x}\right )\right )^2+24 \text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (\frac {b x}{a}+1\right )+24 \text {Li}_3\left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right )+24 \text {Li}_3\left (\frac {b x}{a+b x}\right ) \log \left (-\frac {b x}{a+b x}\right )+24 \text {Li}_3\left (\frac {b x}{a}+1\right ) \left (\log \left (-\frac {b x}{a+b x}\right )-\log (x)\right )+\log ^4\left (-\frac {b x}{a}\right )+\log ^4\left (-\frac {b x}{a+b x}\right )-4 \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {a}{a+b x}\right )\right ) \log ^3\left (-\frac {b x}{a+b x}\right )+6 \log ^2\left (-\frac {b x}{a+b x}\right ) \log ^2\left (-\frac {b x}{a}\right )-4 \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {b x}{a}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (3 \log \left (-\frac {b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac {b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac {b x}{a}+1\right )\right ) \]
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Rubi [F] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log (x) \log ^2(a+b x)}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log (x) \log ^2(a+b x)}{x} \, dx &=\frac {1}{2} \log ^2(x) \log ^2(a+b x)-b \int \frac {\log ^2(x) \log (a+b x)}{a+b x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.10, size = 519, normalized size = 1.00 \[ \frac {1}{12} \left (-24 \left (\text {Li}_4\left (-\frac {b x}{a}\right )+\text {Li}_4\left (\frac {b x}{a+b x}\right )-\text {Li}_4\left (\frac {b x}{a}+1\right )\right )+12 \text {Li}_2\left (-\frac {b x}{a}\right ) \left (\log ^2\left (-\frac {b x}{a}\right )-2 \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (\frac {b x}{a}+1\right )\right ) \log \left (-\frac {b x}{a}\right )+2 \log (x) \left (\log \left (\frac {b x}{a}+1\right )-\log (a+b x)\right )\right )-12 \text {Li}_2\left (\frac {b x}{a+b x}\right ) \log ^2\left (-\frac {b x}{a+b x}\right )+12 \text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {b x}{a+b x}\right )\right )^2+24 \text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (\frac {b x}{a}+1\right )+24 \text {Li}_3\left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right )+24 \text {Li}_3\left (\frac {b x}{a+b x}\right ) \log \left (-\frac {b x}{a+b x}\right )+24 \text {Li}_3\left (\frac {b x}{a}+1\right ) \left (\log \left (-\frac {b x}{a+b x}\right )-\log (x)\right )+\log ^4\left (-\frac {b x}{a}\right )+\log ^4\left (-\frac {b x}{a+b x}\right )-4 \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {a}{a+b x}\right )\right ) \log ^3\left (-\frac {b x}{a+b x}\right )+6 \log ^2\left (-\frac {b x}{a+b x}\right ) \log ^2\left (-\frac {b x}{a}\right )-4 \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (\frac {b x}{a}+1\right )\right ) \log ^2\left (-\frac {b x}{a}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (3 \log \left (-\frac {b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac {b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac {b x}{a}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (b x + a\right )^{2} \log \relax (x)}{x}, x\right ) \]
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 \[ \int \frac {\log \left (b x + a\right )^{2} \log \relax (x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\ln \relax (x ) \ln \left (b x +a \right )^{2}}{x}\, dx \]
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 \[ \frac {1}{2} \, \log \left (b x + a\right )^{2} \log \relax (x)^{2} - b \int \frac {\log \left (b x + a\right ) \log \relax (x)^{2}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (a+b\,x\right )}^2\,\ln \relax (x)}{x} \,d x \]
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 \[ - b \int \frac {\log {\relax (x )}^{2} \log {\left (a + b x \right )}}{a + b x}\, dx + \frac {\log {\relax (x )}^{2} \log {\left (a + b x \right )}^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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