Optimal. Leaf size=225 \[ \frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e^2}+\frac {g 2^{-n-1} e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{e^2} \]
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Rubi [A] time = 0.21, antiderivative size = 225, normalized size of antiderivative = 1.00, 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 2181
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
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*}
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Mathematica [A] time = 0.22, size = 181, normalized size = 0.80 \[ \frac {2^{-n-1} e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^n \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-n} \left (2^{n+1} e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \Gamma \left (n+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \Gamma \left (n+1,-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )}{e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}, x\right ) \]
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 \[ \int {\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.22, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right ) \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{n}\, dx \]
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 = -0.00 \[ \int \left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{n} \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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