Optimal. Leaf size=103 \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e} \]
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Rubi [A]
time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2337,
2212} \begin {gather*} \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2337
Rule 2436
Rubi steps
\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {x}{n}} (a+b x)^p \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\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 )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 103, normalized size = 1.00 \begin {gather*} \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\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 )^p \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{p}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 59, normalized size = 0.57 \begin {gather*} e^{\left (-\frac {b n p \log \left (-\frac {1}{b n}\right ) + b \log \left (c\right ) + a}{b n} - 1\right )} \Gamma \left (p + 1, -\frac {b n \log \left (x e + d\right ) + b \log \left (c\right ) + a}{b n}\right ) \end {gather*}
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 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{p}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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