Optimal. Leaf size=51 \[ -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-3 b n \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 = {2504, 2441,
2352} \begin {gather*} -3 b n \text {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right )-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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x} \, dx &=-\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*}
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Mathematica [A]
time = 0.00, size = 53, normalized size = 1.04 \begin {gather*} -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 \text {Li}_2\left (\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs.
\(2 (47) = 94\).
time = 0.73, size = 183, normalized size = 3.59 \begin {gather*} -3 \, {\left (\log \left (d e^{\left (\frac {1}{3} \, \log \left (x\right ) - 1\right )} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-d e^{\left (\frac {1}{3} \, \log \left (x\right ) - 1\right )}\right )\right )} b n + \frac {1}{12} \, {\left (2 \, b n e^{2} \log \left (x\right )^{2} + 9 \, b d^{2} n x^{\frac {2}{3}} - 36 \, b d n x^{\frac {1}{3}} e + 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 ) + 12 \, {\left (b \log \left (c\right ) + a\right )} e^{2} \log \left (x\right ) - 6 \, {\left (b d^{2} n x^{\frac {2}{3}} - 2 \, b d n x^{\frac {1}{3}} e\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 n x e \log \left (x\right ) - 3 \, b d n x e\right )}}{x^{\frac {2}{3}}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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