Optimal. Leaf size=41 \[ b \text{Unintegrable}\left (\frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )},x\right )+\frac{a \log \left (\log \left (f x^p\right )\right )}{p} \]
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Rubi [A] time = 0.292362, 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 = {} \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \]
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 \left (f x^p\right )} \, dx &=\int \left (\frac{a}{x \log \left (f x^p\right )}+\frac{b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )}\right ) \, dx\\ &=a \int \frac{1}{x \log \left (f x^p\right )} \, dx+b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx\\ &=b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx+\frac{a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log \left (f x^p\right )\right )}{p}\\ &=\frac{a \log \left (\log \left (f x^p\right )\right )}{p}+b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.54297, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d+e{x}^{m} \right ) ^{n} \right ) }{x\ln \left ( f{x}^{p} \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*} b \int \frac{\log \left ({\left (e x^{m} + d\right )}^{n}\right ) + \log \left (c\right )}{x \log \left (f\right ) + x \log \left (x^{p}\right )}\,{d x} + \frac{a \log \left (\log \left (f x^{p}\right )\right )}{p} \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 )}, 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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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