Optimal. Leaf size=45 \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]
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Rubi [A] time = 0.0283106, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2299, 2181} \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2299
Rule 2181
Rubi steps
\begin{align*} \int \log ^p(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^p(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int e^x x^p \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=\frac{\Gamma (1+p,-\log (c (d+e x))) (-\log (c (d+e x)))^{-p} \log ^p(c (d+e x))}{c e}\\ \end{align*}
Mathematica [A] time = 0.0155742, size = 45, normalized size = 1. \[ \frac{(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \text{Gamma}(p+1,-\log (c (d+e x)))}{c e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.224, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25065, size = 72, normalized size = 1.6 \begin{align*} -\frac{\left (-\log \left (c e x + c d\right )\right )^{-p - 1} \log \left (c e x + c d\right )^{p + 1} \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15524, size = 66, normalized size = 1.47 \begin{align*} \frac{\cos \left (\pi p\right ) \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 42.0275, size = 54, normalized size = 1.2 \begin{align*} \begin{cases} \tilde{\infty }^{p} x & \text{for}\: c = 0 \\x \log{\left (c d \right )}^{p} & \text{for}\: e = 0 \\\frac{\left (- \log{\left (c d + c e x \right )}\right )^{- p} \log{\left (c d + c e x \right )}^{p} \Gamma \left (p + 1, - \log{\left (c d + c e x \right )}\right )}{c e} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left ({\left (e x + d\right )} c\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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