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 = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 2295
Rule 2296
Rule 2389
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.84, size = 60, normalized size = 0.98 \[ \frac {{\left (e x + d\right )} \log \left (c e x + c d\right )^{3} - 3 \, {\left (e x + d\right )} \log \left (c e x + c d\right )^{2} - 6 \, e x + 6 \, {\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 71, normalized size = 1.16 \[ {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} - 3 \, {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} + 6 \, {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) - 6 \, {\left (x e + d\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 98, normalized size = 1.61 \[ x \ln \left (c e x +c d \right )^{3}+\frac {d \ln \left (c e x +c d \right )^{3}}{e}-3 x \ln \left (c e x +c d \right )^{2}-\frac {3 d \ln \left (c e x +c d \right )^{2}}{e}+6 x \ln \left (c e x +c d \right )+\frac {6 d \ln \left (c e x +c d \right )}{e}-6 x -\frac {6 d}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 125, normalized size = 2.05 \[ -3 \, e {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )} c\right )^{2} + x \log \left ({\left (e x + d\right )} c\right )^{3} - e {\left (\frac {3 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{2}} - \frac {d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )}{e^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 88, normalized size = 1.44 \[ 6\,x\,\ln \left (c\,d+c\,e\,x\right )-6\,x-3\,x\,{\ln \left (c\,d+c\,e\,x\right )}^2+x\,{\ln \left (c\,d+c\,e\,x\right )}^3-\frac {3\,d\,{\ln \left (c\,d+c\,e\,x\right )}^2}{e}+\frac {d\,{\ln \left (c\,d+c\,e\,x\right )}^3}{e}+\frac {6\,d\,\ln \left (d+e\,x\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 68, normalized size = 1.11 \[ - 6 e \left (- \frac {d \log {\left (d + e x \right )}}{e^{2}} + \frac {x}{e}\right ) + 6 x \log {\left (c \left (d + e x\right ) \right )} + \frac {\left (- 3 d - 3 e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{2}}{e} + \frac {\left (d + e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{3}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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