Optimal. Leaf size=89 \[ 2^{-p-1} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
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Rubi [A] time = 0.0515543, antiderivative size = 89, 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 = {2310, 2181} \[ 2^{-p-1} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{2 x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=2^{-1-p} e^{-\frac{2 a}{b n}} x^2 \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}\\ \end{align*}
Mathematica [A] time = 0.0812065, size = 89, normalized size = 1. \[ 2^{-p-1} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, 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 ({\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x, 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 x \left (a + b \log{\left (c x^{n} \right )}\right )^{p}\, 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{\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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