Optimal. Leaf size=45 \[ \frac {(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \Gamma (p+1,-\log (c (d+e x)))}{c e} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 2181
Rule 2299
Rule 2389
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*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \[ \frac {(-\log (c (d+e x)))^{-p} \log ^p(c (d+e x)) \Gamma (p+1,-\log (c (d+e x)))}{c e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 26, normalized size = 0.58 \[ \frac {\cos \left (\pi p\right ) \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left ({\left (e x + d\right )} c\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \ln \left (\left (e x +d \right ) c \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 53, normalized size = 1.18 \[ -\frac {\left (-\log \left (c e x + c d\right )\right )^{-p - 1} \log \left (c e x + c d\right )^{p + 1} \Gamma \left (p + 1, -\log \left (c e x + c d\right )\right )}{c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 45, normalized size = 1.00 \[ \frac {{\ln \left (c\,\left (d+e\,x\right )\right )}^p\,\Gamma \left (p+1,-\ln \left (c\,\left (d+e\,x\right )\right )\right )}{c\,e\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.94, size = 54, normalized size = 1.20 \[ \begin {cases} \tilde {\infty }^{p} x & \text {for}\: c = 0 \\x \log {\left (c d \right )}^{p} & \text {for}\: e = 0 \\\frac {\left (- \log {\left (c d + c e x \right )}\right )^{- p} \log {\left (c d + c e x \right )}^{p} \Gamma \left (p + 1, - \log {\left (c d + c e x \right )}\right )}{c e} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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