Optimal. Leaf size=107 \[ \frac{\left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p}-\frac{d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p} \]
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Rubi [A] time = 0.147369, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {2454, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{\left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p}-\frac{d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )}{3 e^2 p} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2399
Rule 2389
Rule 2300
Rule 2178
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{x^5}{\log \left (c \left (d+e x^3\right )^p\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{d}{e \log \left (c (d+e x)^p\right )}+\frac{d+e x}{e \log \left (c (d+e x)^p\right )}\right ) \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{d+e x}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e}-\frac{d \operatorname{Subst}\left (\int \frac{1}{\log \left (c (d+e x)^p\right )} \, dx,x,x^3\right )}{3 e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2}-\frac{d \operatorname{Subst}\left (\int \frac{1}{\log \left (c x^p\right )} \, dx,x,d+e x^3\right )}{3 e^2}\\ &=\frac{\left (\left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^2 p}-\frac{\left (d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p}}}{x} \, dx,x,\log \left (c \left (d+e x^3\right )^p\right )\right )}{3 e^2 p}\\ &=-\frac{d \left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-1/p} \text{Ei}\left (\frac{\log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^2 p}+\frac{\left (d+e x^3\right )^2 \left (c \left (d+e x^3\right )^p\right )^{-2/p} \text{Ei}\left (\frac{2 \log \left (c \left (d+e x^3\right )^p\right )}{p}\right )}{3 e^2 p}\\ \end{align*}
Mathematica [A] time = 0.115391, size = 96, normalized size = 0.9 \[ -\frac{\left (d+e x^3\right ) \left (c \left (d+e x^3\right )^p\right )^{-2/p} \left (d \left (c \left (d+e x^3\right )^p\right )^{\frac{1}{p}} \text{Ei}\left (\frac{\log \left (c \left (e x^3+d\right )^p\right )}{p}\right )-\left (d+e x^3\right ) \text{Ei}\left (\frac{2 \log \left (c \left (e x^3+d\right )^p\right )}{p}\right )\right )}{3 e^2 p} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.553, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{\ln \left ( c \left ( e{x}^{3}+d \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{x^{5}}{\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8404, size = 162, normalized size = 1.51 \begin{align*} -\frac{c^{\left (\frac{1}{p}\right )} d \logintegral \left ({\left (e x^{3} + d\right )} c^{\left (\frac{1}{p}\right )}\right ) - \logintegral \left ({\left (e^{2} x^{6} + 2 \, d e x^{3} + d^{2}\right )} c^{\frac{2}{p}}\right )}{3 \, c^{\frac{2}{p}} e^{2} p} \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 = 1.27341, size = 97, normalized size = 0.91 \begin{align*} -\frac{1}{3} \,{\left (\frac{d{\rm Ei}\left (\frac{\log \left (c\right )}{p} + \log \left (x^{3} e + d\right )\right ) e^{\left (-1\right )}}{c^{\left (\frac{1}{p}\right )} p} - \frac{{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{p} + 2 \, \log \left (x^{3} e + d\right )\right ) e^{\left (-1\right )}}{c^{\frac{2}{p}} p}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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