Optimal. Leaf size=51 \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
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Rubi [A] time = 0.0559981, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2310, 2178} \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left (x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{e^{-\frac{4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0577181, size = 51, normalized size = 1. \[ \frac{x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{a+b\ln \left ( c{x}^{n} \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^{3}}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.904, size = 108, normalized size = 2.12 \begin{align*} \frac{e^{\left (-\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left (x^{4} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )}{b 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{x^{3}}{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19458, size = 65, normalized size = 1.27 \begin{align*} \frac{{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac{4 \, a}{b n}\right )}}{b c^{\frac{4}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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