Optimal. Leaf size=225 \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e^2}+\frac{g 2^{-n-1} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^2} \]
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Rubi [A] time = 0.205189, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2401, 2389, 2300, 2181, 2390, 2310} \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e^2}+\frac{g 2^{-n-1} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 2401
Rule 2389
Rule 2300
Rule 2181
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx &=\int \left (\frac{(e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^n}{e}\right ) \, dx\\ &=\frac{g \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e}+\frac{(e f-d g) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^n \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^n \, dx,x,d+e x\right )}{e^2}\\ &=\frac{\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{2 x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}+\frac{\left ((e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{x}{n}} (a+b x)^n \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}\\ &=\frac{2^{-1-n} e^{-\frac{2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \Gamma \left (1+n,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n}}{e^2}+\frac{e^{-\frac{a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+n,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n}}{e^2}\\ \end{align*}
Mathematica [A] time = 0.206539, size = 181, normalized size = 0.8 \[ \frac{2^{-n-1} e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \left (2^{n+1} e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \text{Gamma}\left (n+1,-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )}{e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.957, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{n}\, 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 (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}, 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 \left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{n} \left (f + g x\right )\, 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 (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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