Optimal. Leaf size=42 \[ \frac{(e+f x)^{1-p} (g (e+f x))^{p-1} \text{li}\left (d (e+f x)^p\right )}{d f p} \]
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Rubi [A] time = 0.0760003, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2390, 2308, 2307, 2298} \[ \frac{(e+f x)^{1-p} (g (e+f x))^{p-1} \text{li}\left (d (e+f x)^p\right )}{d f p} \]
Antiderivative was successfully verified.
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Rule 2390
Rule 2308
Rule 2307
Rule 2298
Rubi steps
\begin{align*} \int \frac{(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(g x)^{-1+p}}{\log \left (d x^p\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac{\left ((e+f x)^{1-p} (g (e+f x))^{-1+p}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+p}}{\log \left (d x^p\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac{\left ((e+f x)^{1-p} (g (e+f x))^{-1+p}\right ) \operatorname{Subst}\left (\int \frac{1}{\log (d x)} \, dx,x,(e+f x)^p\right )}{f p}\\ &=\frac{(e+f x)^{1-p} (g (e+f x))^{-1+p} \text{li}\left (d (e+f x)^p\right )}{d f p}\\ \end{align*}
Mathematica [A] time = 0.0217888, size = 42, normalized size = 1. \[ \frac{(e+f x)^{1-p} (g (e+f x))^{p-1} \text{li}\left (d (e+f x)^p\right )}{d f p} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fgx+eg \right ) ^{-1+p}}{\ln \left ( d \left ( fx+e \right ) ^{p} \right ) }}\, 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{{\left (f g x + e g\right )}^{p - 1}}{\log \left ({\left (f x + e\right )}^{p} d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08763, size = 63, normalized size = 1.5 \begin{align*} \frac{g^{p - 1}{\rm Ei}\left (p \log \left (f x + e\right ) + \log \left (d\right )\right )}{d f p} \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{\left (g \left (e + f x\right )\right )^{p - 1}}{\log{\left (d \left (e + f x\right )^{p} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f g x + e g\right )}^{p - 1}}{\log \left ({\left (f x + e\right )}^{p} d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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