Optimal. Leaf size=81 \[ \frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{24 (d+e x) \log (c (d+e x))}{e}+24 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0350419, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{24 (d+e x) \log (c (d+e x))}{e}+24 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2389
Rule 2296
Rule 2295
Rubi steps
\begin{align*} \int \log ^4(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^4(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 \operatorname{Subst}\left (\int \log ^3(c x) \, dx,x,d+e x\right )}{e}\\ &=-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}+\frac{12 \operatorname{Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{24 \operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=24 x-\frac{24 (d+e x) \log (c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.0077568, size = 74, normalized size = 0.91 \[ \frac{(d+e x) \log ^4(c (d+e x))-4 (d+e x) \log ^3(c (d+e x))+12 (d+e x) \log ^2(c (d+e x))-24 (d+e x) \log (c (d+e x))+24 e x}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 129, normalized size = 1.6 \begin{align*} \left ( \ln \left ( cex+cd \right ) \right ) ^{4}x+{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{4}d}{e}}-4\, \left ( \ln \left ( cex+cd \right ) \right ) ^{3}x-4\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{3}d}{e}}+12\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}x+12\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{2}d}{e}}-24\,\ln \left ( cex+cd \right ) x-24\,{\frac{\ln \left ( cex+cd \right ) d}{e}}+24\,x+24\,{\frac{d}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.25579, size = 254, normalized size = 3.14 \begin{align*} -4 \, 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 )^{3} + x \log \left ({\left (e x + d\right )} c\right )^{4} +{\left (e{\left (\frac{4 \,{\left (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 )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{3}} - \frac{d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )}{e^{3}}\right )} - \frac{6 \,{\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 )^{2}}{e^{2}}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84186, size = 192, normalized size = 2.37 \begin{align*} \frac{{\left (e x + d\right )} \log \left (c e x + c d\right )^{4} - 4 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{3} + 12 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{2} + 24 \, e x - 24 \,{\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.62818, size = 88, normalized size = 1.09 \begin{align*} 24 e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) - 24 x \log{\left (c \left (d + e x\right ) \right )} + \frac{\left (- 4 d - 4 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{3}}{e} + \frac{\left (d + e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{4}}{e} + \frac{\left (12 d + 12 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{2}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30065, size = 124, normalized size = 1.53 \begin{align*}{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{4} - 4 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} + 12 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} - 24 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) + 24 \,{\left (x e + d\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]