Optimal. Leaf size=89 \[ -\frac{2^{-p-1} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2} \]
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Rubi [A] time = 0.0583266, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2310, 2181} \[ -\frac{2^{-p-1} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int e^{-\frac{2 x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n x^2}\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2}\\ \end{align*}
Mathematica [A] time = 0.0781012, size = 89, normalized size = 1. \[ -\frac{2^{-p-1} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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^{n}\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^{n} \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^{n}\right ) + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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