Optimal. Leaf size=93 \[ -6 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+6 b^2 n^2 \text{PolyLog}\left (3,\frac{e}{d \sqrt [3]{x}}+1\right )-3 \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \]
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Rubi [A] time = 0.130883, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2454, 2396, 2433, 2374, 6589} \[ -6 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+6 b^2 n^2 \text{PolyLog}\left (3,\frac{e}{d \sqrt [3]{x}}+1\right )-3 \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2396
Rule 2433
Rule 2374
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 )^2}{x} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^2 \log \left (-\frac{e}{d \sqrt [3]{x}}\right )+(6 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 )}{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 )^2 \log \left (-\frac{e}{d \sqrt [3]{x}}\right )+(6 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \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 )^2 \log \left (-\frac{e}{d \sqrt [3]{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt [3]{x}}\right )+\left (6 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\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 )^2 \log \left (-\frac{e}{d \sqrt [3]{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt [3]{x}}\right )+6 b^2 n^2 \text{Li}_3\left (1+\frac{e}{d \sqrt [3]{x}}\right )\\ \end{align*}
Mathematica [B] time = 0.209201, size = 389, normalized size = 4.18 \[ 2 b n \left (3 \text{PolyLog}\left (2,-\frac{e}{d \sqrt [3]{x}}\right )+\log (x) \left (\log \left (d+\frac{e}{\sqrt [3]{x}}\right )-\log \left (\frac{e}{d \sqrt [3]{x}}+1\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )+3 b^2 n^2 \left (\frac{1}{81} \left (-162 \text{PolyLog}\left (3,\frac{d \sqrt [3]{x}}{e}+1\right )-162 \text{PolyLog}\left (3,-\frac{d \sqrt [3]{x}}{e}\right )+9 \log ^2(x) \log \left (d+\frac{e}{\sqrt [3]{x}}\right )-9 \log ^2(x) \log \left (\frac{d \sqrt [3]{x}}{e}+1\right )+27 \log (x) \log ^2\left (d+\frac{e}{\sqrt [3]{x}}\right )-27 \log (x) \log ^2\left (\frac{e}{d}+\sqrt [3]{x}\right )+81 \log ^2\left (\frac{e}{d}+\sqrt [3]{x}\right ) \log \left (-\frac{d \sqrt [3]{x}}{e}\right )-54 \log (x) \log \left (d+\frac{e}{\sqrt [3]{x}}\right ) \log \left (\frac{d \sqrt [3]{x}}{e}+1\right )+54 \log (x) \log \left (\frac{e}{d}+\sqrt [3]{x}\right ) \log \left (\frac{d \sqrt [3]{x}}{e}+1\right )+\log ^3(x)\right )+2 \log \left (\frac{e}{d}+\sqrt [3]{x}\right ) \text{PolyLog}\left (2,\frac{d \sqrt [3]{x}}{e}+1\right )-2 \left (\log \left (d+\frac{e}{\sqrt [3]{x}}\right )-\log \left (\frac{e}{d}+\sqrt [3]{x}\right )\right ) \text{PolyLog}\left (2,-\frac{d \sqrt [3]{x}}{e}\right )\right )+\log (x) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )\right )^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.351, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b^{2} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right )^{2} \log \left (x\right ) - \int -\frac{3 \,{\left (b^{2} d x + b^{2} e x^{\frac{2}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} + 3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x - 2 \,{\left (b^{2} d n x \log \left (x\right ) - 3 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x + 3 \,{\left (b^{2} d x + b^{2} e x^{\frac{2}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right ) - 3 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{2}{3}}\right )} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right ) - 6 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{2}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right ) + 3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{2}{3}}}{3 \,{\left (d x^{2} + e x^{\frac{5}{3}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )^{n}\right ) + a^{2}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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