Optimal. Leaf size=61 \[ \frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{6 (d+e x) \log (c (d+e x))}{e}-6 x \]
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Rubi [A] time = 0.0266686, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{6 (d+e x) \log (c (d+e x))}{e}-6 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^3(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^3(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 \operatorname{Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{(d+e x) \log ^3(c (d+e x))}{e}+\frac{6 \operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=-6 x+\frac{6 (d+e x) \log (c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{(d+e x) \log ^3(c (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.0055952, size = 57, normalized size = 0.93 \[ \frac{(d+e x) \log ^3(c (d+e x))-3 (d+e x) \log ^2(c (d+e x))+6 (d+e x) \log (c (d+e x))-6 e x}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 98, normalized size = 1.6 \begin{align*} \left ( \ln \left ( cex+cd \right ) \right ) ^{3}x+{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{3}d}{e}}-3\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}x-3\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{2}d}{e}}+6\,\ln \left ( cex+cd \right ) x+6\,{\frac{\ln \left ( cex+cd \right ) d}{e}}-6\,x-6\,{\frac{d}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12432, size = 169, normalized size = 2.77 \begin{align*} -3 \, e{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )} c\right )^{2} + x \log \left ({\left (e x + d\right )} c\right )^{3} - e{\left (\frac{3 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{2}} - \frac{d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )}{e^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86053, size = 143, normalized size = 2.34 \begin{align*} \frac{{\left (e x + d\right )} \log \left (c e x + c d\right )^{3} - 3 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{2} - 6 \, e x + 6 \,{\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.456343, size = 68, normalized size = 1.11 \begin{align*} - 6 e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) + 6 x \log{\left (c \left (d + e x\right ) \right )} + \frac{\left (- 3 d - 3 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{2}}{e} + \frac{\left (d + e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{3}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29826, size = 96, normalized size = 1.57 \begin{align*}{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} - 3 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} + 6 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) - 6 \,{\left (x e + d\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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