Optimal. Leaf size=97 \[ -\frac {6 b \text {Li}_2\left (\frac {b}{a x}+1\right ) \log \left (c \left (a+\frac {b}{x}\right )\right )}{a}+\frac {(a x+b) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log \left (-\frac {b}{a x}\right ) \log ^2\left (c \left (a+\frac {b}{x}\right )\right )}{a}+\frac {6 b \text {Li}_3\left (\frac {b}{a x}+1\right )}{a} \]
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Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2453, 2449, 2454, 2396, 2433, 2374, 6589} \[ -\frac {6 b \text {PolyLog}\left (2,\frac {b}{a x}+1\right ) \log \left (c \left (a+\frac {b}{x}\right )\right )}{a}+\frac {6 b \text {PolyLog}\left (3,\frac {b}{a x}+1\right )}{a}+\frac {(a x+b) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log \left (-\frac {b}{a x}\right ) \log ^2\left (c \left (a+\frac {b}{x}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2433
Rule 2449
Rule 2453
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx &=\int \log ^3\left (a c+\frac {b c}{x}\right ) \, dx\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}+\frac {(3 b) \int \frac {\log ^2\left (a c+\frac {b c}{x}\right )}{x} \, dx}{a}\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log ^2(a c+b c x)}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}+\frac {\left (6 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log (a c+b c x)}{a c+b c x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b c}\right )}{a}\right )}{x} \, dx,x,a c+\frac {b c}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {6 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \text {Li}_2\left (1+\frac {b}{a x}\right )}{a}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a c}\right )}{x} \, dx,x,a c+\frac {b c}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {6 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \text {Li}_2\left (1+\frac {b}{a x}\right )}{a}+\frac {6 b \text {Li}_3\left (1+\frac {b}{a x}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 91, normalized size = 0.94 \[ \frac {-6 b \text {Li}_2\left (\frac {b}{a x}+1\right ) \log \left (\frac {c (a x+b)}{x}\right )+\left ((a x+b) \log \left (\frac {c (a x+b)}{x}\right )-3 b \log \left (-\frac {b}{a x}\right )\right ) \log ^2\left (\frac {c (a x+b)}{x}\right )+6 b \text {Li}_3\left (\frac {b}{a x}+1\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left (\frac {a c x + b c}{x}\right )^{3}, x\right ) \]
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 \[ \int \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \ln \left (\frac {\left (a x +b \right ) c}{x}\right )^{3}\, dx \]
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 \[ \frac {{\left (a x + b\right )} \log \left (a x + b\right )^{3} + 3 \, {\left (a x \log \relax (c) - a x \log \relax (x)\right )} \log \left (a x + b\right )^{2}}{a} + \int \frac {a x \log \relax (c)^{3} + b \log \relax (c)^{3} - {\left (a x + b\right )} \log \relax (x)^{3} + 3 \, {\left (a x \log \relax (c) + b \log \relax (c)\right )} \log \relax (x)^{2} + 3 \, {\left ({\left (\log \relax (c)^{2} - 2 \, \log \relax (c)\right )} a x + b \log \relax (c)^{2} + {\left (a x + b\right )} \log \relax (x)^{2} - 2 \, {\left (a x {\left (\log \relax (c) - 1\right )} + b \log \relax (c)\right )} \log \relax (x)\right )} \log \left (a x + b\right ) - 3 \, {\left (a x \log \relax (c)^{2} + b \log \relax (c)^{2}\right )} \log \relax (x)}{a x + b}\,{d x} \]
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 \[ \int {\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )}^3 \,d x \]
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 \[ 3 b \int \frac {\log {\left (a c + \frac {b c}{x} \right )}^{2}}{a x + b}\, dx + x \log {\left (\frac {c \left (a x + b\right )}{x} \right )}^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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