Optimal. Leaf size=55 \[ \frac{(a+b x) \log ^3(a+b x)}{b}-\frac{3 (a+b x) \log ^2(a+b x)}{b}+\frac{6 (a+b x) \log (a+b x)}{b}-6 x \]
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Rubi [A] time = 0.0184441, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2389, 2296, 2295} \[ \frac{(a+b x) \log ^3(a+b x)}{b}-\frac{3 (a+b x) \log ^2(a+b x)}{b}+\frac{6 (a+b x) \log (a+b x)}{b}-6 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^3(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \log ^3(a+b x)}{b}-\frac{3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac{3 (a+b x) \log ^2(a+b x)}{b}+\frac{(a+b x) \log ^3(a+b x)}{b}+\frac{6 \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{b}\\ &=-6 x+\frac{6 (a+b x) \log (a+b x)}{b}-\frac{3 (a+b x) \log ^2(a+b x)}{b}+\frac{(a+b x) \log ^3(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0055469, size = 51, normalized size = 0.93 \[ \frac{(a+b x) \log ^3(a+b x)-3 (a+b x) \log ^2(a+b x)+6 (a+b x) \log (a+b x)-6 b x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 80, normalized size = 1.5 \begin{align*} \left ( \ln \left ( bx+a \right ) \right ) ^{3}x+{\frac{ \left ( \ln \left ( bx+a \right ) \right ) ^{3}a}{b}}-3\, \left ( \ln \left ( bx+a \right ) \right ) ^{2}x-3\,{\frac{ \left ( \ln \left ( bx+a \right ) \right ) ^{2}a}{b}}+6\,\ln \left ( bx+a \right ) x+6\,{\frac{\ln \left ( bx+a \right ) a}{b}}-6\,x-6\,{\frac{a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25495, size = 50, normalized size = 0.91 \begin{align*} \frac{{\left (\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} + 6 \, \log \left (b x + a\right ) - 6\right )}{\left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92681, size = 127, normalized size = 2.31 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right )^{3} - 3 \,{\left (b x + a\right )} \log \left (b x + a\right )^{2} - 6 \, b x + 6 \,{\left (b x + a\right )} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.427488, size = 63, normalized size = 1.15 \begin{align*} - 6 b \left (- \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b}\right ) + 6 x \log{\left (a + b x \right )} + \frac{\left (- 3 a - 3 b x\right ) \log{\left (a + b x \right )}^{2}}{b} + \frac{\left (a + b x\right ) \log{\left (a + b x \right )}^{3}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19527, size = 84, normalized size = 1.53 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right )^{3}}{b} - \frac{3 \,{\left (b x + a\right )} \log \left (b x + a\right )^{2}}{b} + \frac{6 \,{\left (b x + a\right )} \log \left (b x + a\right )}{b} - \frac{6 \,{\left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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