Optimal. Leaf size=63 \[ c^2 \left (-2^{-p-1}\right ) e^{\frac{2 a}{b}} (a+b \log (c x))^p \left (\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 (a+b \log (c x))}{b}\right ) \]
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Rubi [A] time = 0.0548788, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2309, 2181} \[ c^2 \left (-2^{-p-1}\right ) e^{\frac{2 a}{b}} (a+b \log (c x))^p \left (\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 (a+b \log (c x))}{b}\right ) \]
Antiderivative was successfully verified.
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Rule 2309
Rule 2181
Rubi steps
\begin{align*} \int \frac{(a+b \log (c x))^p}{x^3} \, dx &=c^2 \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^p \, dx,x,\log (c x)\right )\\ &=-2^{-1-p} c^2 e^{\frac{2 a}{b}} \Gamma \left (1+p,\frac{2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (\frac{a+b \log (c x)}{b}\right )^{-p}\\ \end{align*}
Mathematica [A] time = 0.0349996, size = 63, normalized size = 1. \[ c^2 \left (-2^{-p-1}\right ) e^{\frac{2 a}{b}} (a+b \log (c x))^p \left (\frac{a+b \log (c x)}{b}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 (a+b \log (c x))}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( cx \right ) \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33496, size = 59, normalized size = 0.94 \begin{align*} -\frac{{\left (b \log \left (c x\right ) + a\right )}^{p + 1} c^{2} e^{\left (\frac{2 \, a}{b}\right )} E_{-p}\left (\frac{2 \,{\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x \right )}\right )^{p}}{x^{3}}\, dx \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 \frac{{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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