Optimal. Leaf size=135 \[ -3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )-18 b^3 n^3 \text {Li}_4\left (1+\frac {e}{d \sqrt [3]{x}}\right ) \]
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Rubi [A]
time = 0.13, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2443,
2481, 2421, 2430, 6724} \begin {gather*} 18 b^2 n^2 \text {PolyLog}\left (3,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-9 b n \text {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-18 b^3 n^3 \text {PolyLog}\left (4,\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 )^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2430
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x} \, dx &=-\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{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 )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+(9 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+(9 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+\left (18 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )-\left (18 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=-3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-9 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )+18 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )-18 b^3 n^3 \text {Li}_4\left (1+\frac {e}{d \sqrt [3]{x}}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(527\) vs. \(2(135)=270\).
time = 0.20, size = 527, normalized size = 3.90 \begin {gather*} \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (\left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (1+\frac {e}{d \sqrt [3]{x}}\right )\right ) \log (x)+3 \text {Li}_2\left (-\frac {e}{d \sqrt [3]{x}}\right )\right )+9 b^2 n^2 \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (2 \log \left (\frac {e}{d}+\sqrt [3]{x}\right ) \text {Li}_2\left (1+\frac {d \sqrt [3]{x}}{e}\right )-2 \left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (\frac {e}{d}+\sqrt [3]{x}\right )\right ) \text {Li}_2\left (-\frac {d \sqrt [3]{x}}{e}\right )+\frac {1}{81} \left (81 \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )+27 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \log (x)-27 \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right ) \log (x)-54 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log (x)+54 \log \left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log (x)+9 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log ^2(x)-9 \log \left (1+\frac {d \sqrt [3]{x}}{e}\right ) \log ^2(x)+\log ^3(x)-162 \text {Li}_3\left (1+\frac {d \sqrt [3]{x}}{e}\right )-162 \text {Li}_3\left (-\frac {d \sqrt [3]{x}}{e}\right )\right )\right )-3 b^3 n^3 \left (\log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+3 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )-6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )+6 \text {Li}_4\left (1+\frac {e}{d \sqrt [3]{x}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{3}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}}{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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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