Optimal. Leaf size=55 \[ \frac{3}{2} b n \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )+\frac{3}{2} \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \]
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Rubi [A] time = 0.0507591, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2394, 2315} \[ \frac{3}{2} b n \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )+\frac{3}{2} \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{x} \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,x^{2/3}\right )\\ &=\frac{3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )-\frac{1}{2} (3 b e n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^{2/3}\right )\\ &=\frac{3}{2} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )+\frac{3}{2} b n \text{Li}_2\left (1+\frac{e x^{2/3}}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.011762, size = 55, normalized size = 1. \[ \frac{3}{2} b \left (n \text{PolyLog}\left (2,\frac{d+e x^{2/3}}{d}\right )+\log \left (-\frac{e x^{2/3}}{d}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+a \log (x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.384, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.64034, size = 154, normalized size = 2.8 \begin{align*} -\frac{3}{2} \,{\left (2 \, \log \left (\frac{e x^{\frac{2}{3}}}{d} + 1\right ) \log \left (x^{\frac{1}{3}}\right ) +{\rm Li}_2\left (-\frac{e x^{\frac{2}{3}}}{d}\right )\right )} b n + \frac{3 \,{\left (2 \, b e n x^{\frac{2}{3}} \log \left (x^{\frac{1}{3}}\right ) - b e n x^{\frac{2}{3}}\right )}}{2 \, d} + \frac{2 \, b d \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right ) \log \left (x\right ) + 2 \,{\left (b d \log \left (c\right ) + a d\right )} \log \left (x\right ) - \frac{2 \, b e n x \log \left (x\right ) - 3 \, b e n x}{x^{\frac{1}{3}}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a}{x}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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