Optimal. Leaf size=114 \[ \frac{x \left (d x^q\right )^m e^{-\frac{a m q+a}{b n}} \left (c x^n\right )^{-\frac{m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
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Rubi [A] time = 0.107144, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {15, 2310, 2181} \[ \frac{x \left (d x^q\right )^m e^{-\frac{a m q+a}{b n}} \left (c x^n\right )^{-\frac{m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\left (x^{-m q} \left (d x^q\right )^m\right ) \int x^{m q} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1+m q}{n}} \left (d x^q\right )^m\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m q) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{e^{-\frac{a+a m q}{b n}} x \left (c x^n\right )^{-\frac{1+m q}{n}} \left (d x^q\right )^m \Gamma \left (1+p,-\frac{(1+m q) \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{(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q}\\ \end{align*}
Mathematica [A] time = 0.151183, size = 118, normalized size = 1.04 \[ \frac{x^{-m q} \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \exp \left (-\frac{(m q+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.508, size = 0, normalized size = 0. \begin{align*} \int \left ( d{x}^{q} \right ) ^{m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x^{q}\right )^{m}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \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 (d x^{q}\right )^{m}{\left (b \log \left (c x^{n}\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 (d x^{q}\right )^{m}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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