Optimal. Leaf size=139 \[ x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \left (\frac{b d}{m}-\frac{a e}{n q}\right ) \text{Gamma}\left (q+1,-\frac{m \log \left (c x^n\right )}{n}\right )+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac{a x^{2 m} (a e m-b d n q)}{2 b m n q} \]
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Rubi [A] time = 0.170908, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2545, 14, 2310, 2181} \[ x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \left (\frac{b d}{m}-\frac{a e}{n q}\right ) \text{Gamma}\left (q+1,-\frac{m \log \left (c x^n\right )}{n}\right )+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac{a x^{2 m} (a e m-b d n q)}{2 b m n q} \]
Antiderivative was successfully verified.
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Rule 2545
Rule 14
Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int \frac{\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right ) \, dx\\ &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int \left (a x^{-1+2 m}+b x^{-1+m} \log ^q\left (c x^n\right )\right ) \, dx\\ &=\frac{a \left (d-\frac{a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\left (b \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^q\left (c x^n\right ) \, dx\\ &=\frac{a \left (d-\frac{a e m}{b n q}\right ) x^{2 m}}{2 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}-\frac{\left (b \left (-d+\frac{a e m}{b n q}\right ) x^m \left (c x^n\right )^{-\frac{m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{a \left (d-\frac{a e m}{b n q}\right ) x^{2 m}}{2 m}+\left (\frac{b d}{m}-\frac{a e}{n q}\right ) x^m \left (c x^n\right )^{-\frac{m}{n}} \Gamma \left (1+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}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{2 b n q}\\ \end{align*}
Mathematica [A] time = 0.385985, size = 157, normalized size = 1.13 \[ \frac{\left (c x^n\right )^{-\frac{m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} \left (-2 a e m q x^m \log ^q\left (c x^n\right ) \text{Gamma}\left (q,-\frac{m \log \left (c x^n\right )}{n}\right )+2 b d n q x^m \log ^q\left (c x^n\right ) \text{Gamma}\left (q+1,-\frac{m \log \left (c x^n\right )}{n}\right )+\left (c x^n\right )^{m/n} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (a d n q x^{2 m}+b e m \log ^{2 q}\left (c x^n\right )\right )\right )}{2 m n q} \]
Antiderivative was successfully verified.
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Maple [F] time = 10.297, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{m}+e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \right ) \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: ValueError} \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 e x^{m} \log \left (c x^{n}\right )^{q - 1} + a d x^{2 \, m} +{\left (b d x^{m} + b e \log \left (c x^{n}\right )^{q - 1}\right )} \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 )}{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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