Optimal. Leaf size=68 \[ \frac{b e m n \text{Unintegrable}\left (\frac{x^{m-1}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )},x\right )}{2 p}-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )} \]
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Rubi [A] time = 0.116714, 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 = {} \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx &=-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )}+\frac{(b e m n) \int \frac{x^{-1+m}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )} \, dx}{2 p}\\ \end{align*}
Mathematica [A] time = 10.6062, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.468, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d+e{x}^{m} \right ) ^{n} \right ) }{x \left ( \ln \left ( f{x}^{p} \right ) \right ) ^{3}}}\, 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{1}{2} \,{\left (2 \, d e m^{2} n \int \frac{x^{m}}{2 \,{\left (e^{2} p^{2} x x^{2 \, m} \log \left (f\right ) + 2 \, d e p^{2} x x^{m} \log \left (f\right ) + d^{2} p^{2} x \log \left (f\right ) +{\left (e^{2} p^{2} x x^{2 \, m} + 2 \, d e p^{2} x x^{m} + d^{2} p^{2} x\right )} \log \left (x^{p}\right )\right )}}\,{d x} - \frac{e m n x^{m} \log \left (x^{p}\right ) + d p \log \left (c\right ) +{\left (e m n \log \left (f\right ) + e p \log \left (c\right )\right )} x^{m} +{\left (e p x^{m} + d p\right )} \log \left ({\left (e x^{m} + d\right )}^{n}\right )}{e p^{2} x^{m} \log \left (f\right )^{2} + d p^{2} \log \left (f\right )^{2} +{\left (e p^{2} x^{m} + d p^{2}\right )} \log \left (x^{p}\right )^{2} + 2 \,{\left (e p^{2} x^{m} \log \left (f\right ) + d p^{2} \log \left (f\right )\right )} \log \left (x^{p}\right )}\right )} b - \frac{a}{2 \, p \log \left (f x^{p}\right )^{2}} \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{b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}}, 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{b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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