Optimal. Leaf size=65 \[ \frac{e^{-\frac{a}{b}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b}\right )}{c n} \]
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Rubi [A] time = 0.0512487, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2310, 2181} \[ \frac{e^{-\frac{a}{b}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b}\right )}{c n} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int x^{-1+n} \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{c n}\\ &=\frac{e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c x^n\right )}{b}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b}\right )^{-p}}{c n}\\ \end{align*}
Mathematica [A] time = 0.0236361, size = 65, normalized size = 1. \[ \frac{e^{-\frac{a}{b}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b}\right )}{c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.768, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1+n} \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^{n - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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