Optimal. Leaf size=24 \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]
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Rubi [A] time = 0.0086375, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2295} \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \log \left (c (d+e x)^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-n x+\frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0067945, size = 24, normalized size = 1. \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 30, normalized size = 1.3 \begin{align*} \ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-nx+{\frac{dn\ln \left ( ex+d \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14588, size = 47, normalized size = 1.96 \begin{align*} -e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + x \log \left ({\left (e x + d\right )}^{n} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01158, size = 73, normalized size = 3.04 \begin{align*} -\frac{e n x - e x \log \left (c\right ) -{\left (e n x + d n\right )} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.442061, size = 37, normalized size = 1.54 \begin{align*} \begin{cases} \frac{d n \log{\left (d + e x \right )}}{e} + n x \log{\left (d + e x \right )} - n x + x \log{\left (c \right )} & \text{for}\: e \neq 0 \\x \log{\left (c d^{n} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24072, size = 54, normalized size = 2.25 \begin{align*}{\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) -{\left (x e + d\right )} n e^{\left (-1\right )} +{\left (x e + d\right )} e^{\left (-1\right )} \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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