Optimal. Leaf size=309 \[ \frac{4 i b^2 e^{3/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}+\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{d^{3/2}}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.444566, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {2451, 2457, 2471, 2448, 263, 205, 2470, 12, 260, 6688, 4924, 4868, 2447} \[ \frac{4 i b^2 e^{3/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}+\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{d^{3/2}}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2451
Rule 2457
Rule 2471
Rule 2448
Rule 263
Rule 205
Rule 2470
Rule 12
Rule 260
Rule 6688
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+(4 b e n) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d}-\frac{e \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{(4 b e n) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (4 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (4 b^2 e n\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (8 b^2 e^{5/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (8 b^2 e^{5/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}+\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{d^{3/2}}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{\left (8 i b^2 e^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{d^{3/2}}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}+\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac{4 a b e n \sqrt [3]{x}}{d}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{d^{3/2}}+\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac{4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{4 i b^2 e^{3/2} n^2 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.13613, size = 523, normalized size = 1.69 \[ b e n \left (\frac{b \sqrt{e} n \left (-4 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )}{(-d)^{3/2}}+\frac{b d \sqrt{e} n \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )+\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )-4 \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )}{(-d)^{5/2}}-\frac{2 \sqrt{e} \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{(-d)^{3/2}}+\frac{2 \sqrt{e} \log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{(-d)^{3/2}}+\frac{4 a \sqrt [3]{x}}{d}+\frac{4 b \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d}-\frac{8 b \sqrt{e} n \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )}{d^{3/2}}\right )+x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.356, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac{d x + e x^{\frac{1}{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(-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{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{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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