Optimal. Leaf size=74 \[ \left (d-\frac{a e m}{b n q}\right ) \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2},x\right )-\frac{e}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )} \]
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Rubi [A] time = 0.282174, 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 = {} \[ \int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx &=-\frac{e}{b n q \left (a x^m+b \log ^q\left (c x^n\right )\right )}-\left (-d+\frac{a e m}{b n q}\right ) \int \frac{x^{-1+m}}{\left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 7.30777, size = 0, normalized size = 0. \[ \int \frac{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 7.905, size = 0, normalized size = 0. \begin{align*} \int{\frac{d{x}^{m}+e \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 ) ^{2}}}\, 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*} -\frac{b d \log \left (c\right ) + b d \log \left (x^{n}\right ) - a e}{a^{2} b m x^{m} \log \left (x^{n}\right ) -{\left (n q - m \log \left (c\right )\right )} a^{2} b x^{m} +{\left (a b^{2} m \log \left (x^{n}\right ) -{\left (n q - m \log \left (c\right )\right )} a b^{2}\right )}{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} + \int -\frac{{\left (e m n{\left (q - 1\right )} - e m^{2} \log \left (c\right )\right )} a +{\left (d m n q \log \left (c\right ) -{\left (q^{2} - q\right )} d n^{2}\right )} b +{\left (b d m n q - a e m^{2}\right )} \log \left (x^{n}\right )}{a^{2} b m^{2} x x^{m} \log \left (x^{n}\right )^{2} - 2 \,{\left (m n q - m^{2} \log \left (c\right )\right )} a^{2} b x x^{m} \log \left (x^{n}\right ) +{\left (n^{2} q^{2} - 2 \, m n q \log \left (c\right ) + m^{2} \log \left (c\right )^{2}\right )} a^{2} b x x^{m} +{\left (a b^{2} m^{2} x \log \left (x^{n}\right )^{2} - 2 \,{\left (m n q - m^{2} \log \left (c\right )\right )} a b^{2} x \log \left (x^{n}\right ) +{\left (n^{2} q^{2} - 2 \, m n q \log \left (c\right ) + m^{2} \log \left (c\right )^{2}\right )} a b^{2} x\right )}{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} \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{d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{2 \, a b x x^{m} \log \left (c x^{n}\right )^{q} + a^{2} x x^{2 \, m} + b^{2} x \log \left (c x^{n}\right )^{2 \, 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{d x^{m} + e \log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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