Optimal. Leaf size=83 \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]
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Rubi [A] time = 0.131599, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2300, 2178, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2300
Rule 2178
Rule 2445
Rubi steps
\begin{align*} \int \frac{1}{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^n (e+f x)^{m n}\right )} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^n x^{m n}\right )} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{m n}}}{a+b x} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{e^{-\frac{a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n}\\ \end{align*}
Mathematica [A] time = 0.111027, size = 83, normalized size = 1. \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{b f m n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{-1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21535, size = 157, normalized size = 1.89 \begin{align*} \frac{e^{\left (-\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} \logintegral \left ({\left (f x + e\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )}{b f m n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \log{\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19801, size = 107, normalized size = 1.29 \begin{align*} \frac{{\rm Ei}\left (\frac{\log \left (d\right )}{m} + \frac{\log \left (c\right )}{m n} + \frac{a}{b m n} + \log \left (f x + e\right )\right ) e^{\left (-\frac{a}{b m n}\right )}}{b c^{\frac{1}{m n}} d^{\left (\frac{1}{m}\right )} f m n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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