Optimal. Leaf size=135 \[ 18 b^2 n^2 \text {Li}_3\left (\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-9 b n \text {Li}_2\left (\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-18 b^3 n^3 \text {Li}_4\left (\frac {e}{d \sqrt [3]{x}}+1\right ) \]
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Rubi [A] time = 0.20, 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 = {2454, 2396, 2433, 2374, 2383, 6589} \[ 18 b^2 n^2 \text {PolyLog}\left (3,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-9 b n \text {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-18 b^3 n^3 \text {PolyLog}\left (4,\frac {e}{d \sqrt [3]{x}}+1\right )-3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2433
Rule 2454
Rule 6589
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 \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.32, size = 527, normalized size = 3.90 \[ 9 b^2 n^2 \left (\frac {1}{81} \left (-162 \text {Li}_3\left (\frac {\sqrt [3]{x} d}{e}+1\right )-162 \text {Li}_3\left (-\frac {d \sqrt [3]{x}}{e}\right )+9 \log ^2(x) \log \left (d+\frac {e}{\sqrt [3]{x}}\right )-9 \log ^2(x) \log \left (\frac {d \sqrt [3]{x}}{e}+1\right )+27 \log (x) \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )-27 \log (x) \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right )+81 \log ^2\left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )-54 \log (x) \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (\frac {d \sqrt [3]{x}}{e}+1\right )+54 \log (x) \log \left (\frac {e}{d}+\sqrt [3]{x}\right ) \log \left (\frac {d \sqrt [3]{x}}{e}+1\right )+\log ^3(x)\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{x} d}{e}+1\right ) \log \left (\frac {e}{d}+\sqrt [3]{x}\right )-2 \text {Li}_2\left (-\frac {d \sqrt [3]{x}}{e}\right ) \left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (\frac {e}{d}+\sqrt [3]{x}\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )+3 b n \left (3 \text {Li}_2\left (-\frac {e}{d \sqrt [3]{x}}\right )+\log (x) \left (\log \left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (\frac {e}{d \sqrt [3]{x}}+1\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )^2+\log (x) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )^3-3 b^3 n^3 \left (6 \text {Li}_4\left (\frac {e}{d \sqrt [3]{x}}+1\right )+3 \text {Li}_2\left (\frac {e}{d \sqrt [3]{x}}+1\right ) \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )-6 \text {Li}_3\left (\frac {e}{d \sqrt [3]{x}}+1\right ) \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+\log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{n}\right ) + a^{3}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )+a \right )^{3}}{x}\, dx \]
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 \[ b^{3} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right )^{3} \log \relax (x) - \int \frac {{\left (b^{3} d x + b^{3} e x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} + {\left (b^{3} d n x \log \relax (x) - 3 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + 3 \, {\left (b^{3} d x + b^{3} e x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) - 3 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {2}{3}}\right )} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right )^{2} - 3 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x - 3 \, {\left ({\left (b^{3} d x + b^{3} e x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x - 2 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {2}{3}}\right )} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right ) + 3 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) - {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{\frac {2}{3}}}{d x^{2} + e x^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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