Integrand size = 22, antiderivative size = 51 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-3 b n \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 51, 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} \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-3 b n \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right ) \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+(3 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-3 b n \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+a \log (x)-3 b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \]
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\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x}d x\]
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\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (44) = 88\).
Time = 0.46 (sec) , antiderivative size = 185, normalized size of antiderivative = 3.63 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=-3 \, {\left (\log \left (\frac {d x^{\frac {1}{3}}}{e} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {d x^{\frac {1}{3}}}{e}\right )\right )} b n + \frac {2 \, b e^{2} n \log \left (x\right )^{2} + 12 \, b e^{2} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right ) \log \left (x\right ) - 12 \, b e^{2} \log \left (x\right ) \log \left (x^{\frac {1}{3} \, n}\right ) + 9 \, b d^{2} n x^{\frac {2}{3}} - 36 \, b d e n x^{\frac {1}{3}} - 6 \, {\left (b d^{2} n x^{\frac {2}{3}} - 2 \, b d e n x^{\frac {1}{3}}\right )} \log \left (x\right ) + 12 \, {\left (b e^{2} \log \left (c\right ) + a e^{2}\right )} \log \left (x\right ) + \frac {3 \, {\left (2 \, b d^{2} n x \log \left (x\right ) - 3 \, b d^{2} n x\right )}}{x^{\frac {1}{3}}} - \frac {12 \, {\left (b d e n x \log \left (x\right ) - 3 \, b d e n x\right )}}{x^{\frac {2}{3}}}}{12 \, e^{2}} \]
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\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a}{x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x} \,d x \]
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