Integrand size = 22, antiderivative size = 55 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=-\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-\frac {3}{2} b n \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2441, 2352} \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=-\frac {3}{2} \log \left (-\frac {e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-\frac {3}{2} b n \operatorname {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right ) \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {3}{2} \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\frac {1}{x^{2/3}}\right )\right ) \\ & = -\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{x^{2/3}}\right ) \\ & = -\frac {3}{2} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-\frac {3}{2} b n \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=a \log (x)-\frac {3}{2} b \left (\log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right ) \]
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\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x}d x\]
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\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (44) = 88\).
Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.31 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=-\frac {3}{2} \, {\left (2 \, \log \left (\frac {d x^{\frac {2}{3}}}{e} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {d x^{\frac {2}{3}}}{e}\right )\right )} b n + \frac {2 \, b e n \log \left (x\right )^{2} + 6 \, b d n x^{\frac {2}{3}} \log \left (x\right ) + 6 \, b e \log \left ({\left (d x^{\frac {2}{3}} + e\right )}^{n}\right ) \log \left (x\right ) - 12 \, b e \log \left (x\right ) \log \left (x^{\frac {1}{3} \, n}\right ) - 9 \, b d n x^{\frac {2}{3}} + 6 \, {\left (b e \log \left (c\right ) + a e\right )} \log \left (x\right ) - \frac {3 \, {\left (2 \, b d n x \log \left (x\right ) - 3 \, b d n x\right )}}{x^{\frac {1}{3}}}}{6 \, e} \]
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\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x} \,d x \]
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