Optimal. Leaf size=134 \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{e (q+1)} \]
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Rubi [A] time = 0.190116, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2310, 2181, 2445} \[ \frac{(e x)^{q+1} e^{-\frac{a (q+1)}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac{q+1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{e (q+1)} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rule 2445
Rubi steps
\begin{align*} \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx &=\operatorname{Subst}\left (\int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right )^p \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\left ((e x)^{1+q} \left (c d^n x^{m n}\right )^{-\frac{1+q}{m n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+q) x}{m n}} (a+b x)^p \, dx,x,\log \left (c d^n x^{m n}\right )\right )}{e m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac{e^{-\frac{a (1+q)}{b m n}} (e x)^{1+q} \left (c \left (d x^m\right )^n\right )^{-\frac{1+q}{m n}} \Gamma \left (1+p,-\frac{(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac{(1+q) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p}}{e (1+q)}\\ \end{align*}
Mathematica [A] time = 0.193908, size = 133, normalized size = 0.99 \[ \frac{x^{-q} (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \exp \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )-b m n \log (x)\right )}{b m n}\right ) \left (-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(q+1) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{q+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{q} \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{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 (e x\right )^{q}{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}, 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 (e x\right )^{q}{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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