Optimal. Leaf size=67 \[ \frac{\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{a x^m+b \log ^q\left (c x^n\right )},x\right )}{b n q} \]
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Rubi [A] time = 0.244673, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx &=\frac{\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac{(a m) \int \frac{x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )} \, dx}{b n q}\\ \end{align*}
Mathematica [A] time = 0.126489, size = 0, normalized size = 0. \[ \int \frac{\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 8.273, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q}}{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -a \int \frac{x^{m}}{a b x x^{m} \log \left (c\right ) + a b x x^{m} \log \left (x^{n}\right ) +{\left (b^{2} x \log \left (c\right ) + b^{2} x \log \left (x^{n}\right )\right )}{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} + \frac{\log \left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c x^{n}\right )^{q - 1}}{a x x^{m} + b x \log \left (c x^{n}\right )^{q}}, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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