Optimal. Leaf size=519 \[ \frac{1}{12} \left (-24 \left (\text{PolyLog}\left (4,-\frac{b x}{a}\right )+\text{PolyLog}\left (4,\frac{b x}{a+b x}\right )-\text{PolyLog}\left (4,\frac{b x}{a}+1\right )\right )+12 \left (\log ^2\left (-\frac{b x}{a}\right )-2 \left (\log \left (-\frac{b x}{a+b x}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log \left (-\frac{b x}{a}\right )+2 \log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log (a+b x)\right )\right ) \text{PolyLog}\left (2,-\frac{b x}{a}\right )-12 \log ^2\left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (2,\frac{b x}{a+b x}\right )+12 \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{b x}{a+b x}\right )\right )^2 \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{b x}{a}+1\right ) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )+\log (a+b x)\right ) \text{PolyLog}\left (3,-\frac{b x}{a}\right )+24 \log \left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (3,\frac{b x}{a+b x}\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )-\log (x)\right ) \text{PolyLog}\left (3,\frac{b x}{a}+1\right )+\log ^4\left (-\frac{b x}{a}\right )+6 \log ^2\left (-\frac{b x}{a+b x}\right ) \log ^2\left (-\frac{b x}{a}\right )-4 \log \left (-\frac{b x}{a+b x}\right ) \left (\log \left (-\frac{b x}{a}\right )+3 \log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{b x}{a}\right )+\log ^4\left (-\frac{b x}{a+b x}\right )-4 \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{a}{a+b x}\right )\right ) \log ^3\left (-\frac{b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \left (3 \log \left (-\frac{b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac{b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac{b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac{b x}{a}+1\right )\right ) \]
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Rubi [F] time = 0.048933, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.0948574, size = 519, normalized size = 1. \[ \frac{1}{12} \left (-24 \left (\text{PolyLog}\left (4,-\frac{b x}{a}\right )+\text{PolyLog}\left (4,\frac{b x}{a+b x}\right )-\text{PolyLog}\left (4,\frac{b x}{a}+1\right )\right )+12 \left (\log ^2\left (-\frac{b x}{a}\right )-2 \left (\log \left (-\frac{b x}{a+b x}\right )+\log \left (\frac{b x}{a}+1\right )\right ) \log \left (-\frac{b x}{a}\right )+2 \log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log (a+b x)\right )\right ) \text{PolyLog}\left (2,-\frac{b x}{a}\right )-12 \log ^2\left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (2,\frac{b x}{a+b x}\right )+12 \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{b x}{a+b x}\right )\right )^2 \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{b x}{a}+1\right ) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )+\log (a+b x)\right ) \text{PolyLog}\left (3,-\frac{b x}{a}\right )+24 \log \left (-\frac{b x}{a+b x}\right ) \text{PolyLog}\left (3,\frac{b x}{a+b x}\right )+24 \left (\log \left (-\frac{b x}{a+b x}\right )-\log (x)\right ) \text{PolyLog}\left (3,\frac{b x}{a}+1\right )+\log ^4\left (-\frac{b x}{a}\right )+6 \log ^2\left (-\frac{b x}{a+b x}\right ) \log ^2\left (-\frac{b x}{a}\right )-4 \log \left (-\frac{b x}{a+b x}\right ) \left (\log \left (-\frac{b x}{a}\right )+3 \log \left (\frac{b x}{a}+1\right )\right ) \log ^2\left (-\frac{b x}{a}\right )+\log ^4\left (-\frac{b x}{a+b x}\right )-4 \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{a}{a+b x}\right )\right ) \log ^3\left (-\frac{b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+6 \left (\log (x)-\log \left (-\frac{b x}{a}\right )\right ) \left (3 \log \left (-\frac{b x}{a}\right )+\log (x)\right ) \log ^2\left (\frac{b x}{a}+1\right )+4 \left (2 \log ^3\left (-\frac{b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (\frac{b x}{a}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( x \right ) \left ( \ln \left ( bx+a \right ) \right ) ^{2}}{x}}\, 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{1}{2} \, \log \left (b x + a\right )^{2} \log \left (x\right )^{2} - b \int \frac{\log \left (b x + a\right ) \log \left (x\right )^{2}}{b x + a}\,{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 (\frac{\log \left (b x + a\right )^{2} \log \left (x\right )}{x}, 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*} - b \int \frac{\log{\left (x \right )}^{2} \log{\left (a + b x \right )}}{a + b x}\, dx + \frac{\log{\left (x \right )}^{2} \log{\left (a + b x \right )}^{2}}{2} \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 \frac{\log \left (b x + a\right )^{2} \log \left (x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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