Optimal. Leaf size=81 \[ \frac{b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac{a x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \text{Gamma}\left (q,-\frac{m \log \left (c x^n\right )}{n}\right )}{n} \]
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Rubi [A] time = 0.157463, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2539, 2310, 2181, 2302, 30} \[ \frac{b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac{a x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \text{Gamma}\left (q,-\frac{m \log \left (c x^n\right )}{n}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2539
Rule 2310
Rule 2181
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\int \left (a x^{-1+m} \log ^{-1+q}\left (c x^n\right )+\frac{b \log ^{-1+2 q}\left (c x^n\right )}{x}\right ) \, dx\\ &=a \int x^{-1+m} \log ^{-1+q}\left (c x^n\right ) \, dx+b \int \frac{\log ^{-1+2 q}\left (c x^n\right )}{x} \, dx\\ &=\frac{b \operatorname{Subst}\left (\int x^{-1+2 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac{\left (a x^m \left (c x^n\right )^{-\frac{m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{m x}{n}} x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac{a x^m \left (c x^n\right )^{-\frac{m}{n}} \Gamma \left (q,-\frac{m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q}}{n}\\ \end{align*}
Mathematica [A] time = 0.155532, size = 77, normalized size = 0.95 \[ \frac{\log ^q\left (c x^n\right ) \left (\frac{b \log ^q\left (c x^n\right )}{q}-2 a x^m \left (c x^n\right )^{-\frac{m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \text{Gamma}\left (q,-\frac{m \log \left (c x^n\right )}{n}\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.949, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) }{x}}\, 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 (\frac{a x^{m} \log \left (c x^{n}\right )^{q - 1} + b \log \left (c x^{n}\right )^{q - 1} \log \left (c x^{n}\right )^{q}}{x}, 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 \frac{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} \log \left (c x^{n}\right )^{q - 1}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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