Optimal. Leaf size=51 \[ 3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+3 b n \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right ) \]
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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 \sqrt [3]{x}}{d}+1\right )+3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+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+e \sqrt [3]{x}\right )^n\right )}{x} \, dx &=3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )-(3 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+3 b n \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\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+e \sqrt [3]{x}\right )^n\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+a \log (x)+3 b n \text {Li}_2\left (\frac {d+e \sqrt [3]{x}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x}\, dx\]
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 = 0.44, size = 183, normalized size = 3.59 \begin {gather*} -3 \, {\left (\log \left (x^{\frac {1}{3}}\right ) \log \left (\frac {e^{\left (\frac {1}{3} \, \log \left (x\right ) + 1\right )}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e^{\left (\frac {1}{3} \, \log \left (x\right ) + 1\right )}}{d}\right )\right )} b n + \frac {4 \, b d^{2} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n}\right ) \log \left (x\right ) + 4 \, {\left (b d^{2} \log \left (c\right ) + a d^{2}\right )} \log \left (x\right ) + \frac {2 \, b n x e^{2} \log \left (x\right ) - 3 \, b n x e^{2}}{x^{\frac {1}{3}}} - \frac {4 \, {\left (b d n x e \log \left (x\right ) - 3 \, b d n x e\right )}}{x^{\frac {2}{3}}}}{4 \, d^{2}} - \frac {3 \, {\left (4 \, b d n e^{\left (\frac {1}{3} \, \log \left (x\right ) + 1\right )} - b n e^{\left (\frac {2}{3} \, \log \left (x\right ) + 2\right )} - 2 \, {\left (2 \, b d n e^{\left (\frac {1}{3} \, \log \left (x\right ) + 1\right )} - b n e^{\left (\frac {2}{3} \, \log \left (x\right ) + 2\right )}\right )} \log \left (x^{\frac {1}{3}}\right )\right )}}{4 \, d^{2}} \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 + 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+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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