Optimal. Leaf size=227 \[ \frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right )}{d^3}-\frac{2 b e^3 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac{2 b e^2 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]
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Rubi [A] time = 0.53316, antiderivative size = 248, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {2451, 2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right )}{d^3}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}-\frac{2 b e^3 n \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac{2 b e^2 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 2451
Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d}+\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d}\\ &=\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^2}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d}\\ &=-\frac{2 b e^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d}+\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac{b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{2 b e^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{2 b e^3 n \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 )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3}+\frac{\left (2 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac{b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac{b^2 e^3 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{2 b e^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac{b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{2 b e^3 n \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 )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3}-\frac{2 b^2 e^3 n^2 \text{Li}_2\left (1+\frac{e}{d \sqrt [3]{x}}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.18659, size = 237, normalized size = 1.04 \[ x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{b e n \left (3 b e^2 n \left (\log \left (d \sqrt [3]{x}+e\right ) \left (\log \left (d \sqrt [3]{x}+e\right )-2 \log \left (-\frac{d \sqrt [3]{x}}{e}\right )\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt [3]{x}}{e}+1\right )\right )-3 d^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-6 e^2 \log \left (d \sqrt [3]{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+6 a d e \sqrt [3]{x}+6 b d e \sqrt [3]{x} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+2 b e^2 n \left (3 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )+\log (x)\right )+3 b e n \left (e \log \left (d \sqrt [3]{x}+e\right )-d \sqrt [3]{x}\right )\right )}{3 d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.341, size = 0, normalized size = 0. \begin{align*} \int \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*}{\left (e n{\left (\frac{2 \, e^{2} \log \left (d x^{\frac{1}{3}} + e\right )}{d^{3}} + \frac{d x^{\frac{2}{3}} - 2 \, e x^{\frac{1}{3}}}{d^{2}}\right )} + 2 \, x \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )\right )} a b +{\left (x \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right )^{2} - \int -\frac{3 \, d x \log \left (c\right )^{2} + 3 \, e x^{\frac{2}{3}} \log \left (c\right )^{2} + 3 \,{\left (d x + e x^{\frac{2}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} - 2 \,{\left (d n x - 3 \, d x \log \left (c\right ) - 3 \, e x^{\frac{2}{3}} \log \left (c\right ) + 3 \,{\left (d x + e x^{\frac{2}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right ) - 6 \,{\left (d x \log \left (c\right ) + e x^{\frac{2}{3}} \log \left (c\right )\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{3 \,{\left (d x + e x^{\frac{2}{3}}\right )}}\,{d x}\right )} b^{2} + a^{2} 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 (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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d + \frac{e}{\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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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