Optimal. Leaf size=89 \[ 2^{-p-1} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 2181
Rule 2310
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*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 1.00 \[ 2^{-p-1} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x, x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int x \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \log {\left (c x^{n} \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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