Optimal. Leaf size=42 \[ \frac {(e+f x)^{1-p} (g (e+f x))^{-1+p} \text {li}\left (d (e+f x)^p\right )}{d f p} \]
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Rubi [A]
time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2437, 2345,
2344, 2335} \begin {gather*} \frac {(e+f x)^{1-p} (g (e+f x))^{p-1} \text {li}\left (d (e+f x)^p\right )}{d f p} \end {gather*}
Antiderivative was successfully verified.
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Rule 2335
Rule 2344
Rule 2345
Rule 2437
Rubi steps
\begin {align*} \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx &=\frac {\text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 1.02 \begin {gather*} \frac {(e+f x)^{1-p} (g (e+f x))^{-1+p} \text {Ei}\left (\log \left (d (e+f x)^p\right )\right )}{d f p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (f g x +e g \right )^{-1+p}}{\ln \left (d \left (f x +e \right )^{p}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 28, normalized size = 0.67 \begin {gather*} \frac {g^{p - 1} {\rm Ei}\left (p \log \left (f x + e\right ) + \log \left (d\right )\right )}{d f p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (g \left (e + f x\right )\right )^{p - 1}}{\log {\left (d \left (e + f x\right )^{p} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (e\,g+f\,g\,x\right )}^{p-1}}{\ln \left (d\,{\left (e+f\,x\right )}^p\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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