Optimal. Leaf size=276 \[ -\frac{b d^3 n \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^3}+\frac{3 b d^2 n \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac{3 b d n \left (d+\frac{e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac{b n \left (d+\frac{e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac{3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac{b^2 d^3 n^2 \log ^2\left (d+\frac{e}{x^{2/3}}\right )}{2 e^3}+\frac{3 b^2 d n^2 \left (d+\frac{e}{x^{2/3}}\right )^2}{4 e^3}-\frac{b^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^3}{9 e^3} \]
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Rubi [A] time = 0.300981, antiderivative size = 217, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac{3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac{b^2 d^3 n^2 \log ^2\left (d+\frac{e}{x^{2/3}}\right )}{2 e^3}+\frac{3 b^2 d n^2 \left (d+\frac{e}{x^{2/3}}\right )^2}{4 e^3}-\frac{b^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^3}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b e n) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )\\ &=\frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+\frac{e}{x^{2/3}}\right )\\ &=\frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{6 e^3}\\ &=\frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac{6 d^3 \log (x)}{x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{6 e^3}\\ &=\frac{3 b^2 d n^2 \left (d+\frac{e}{x^{2/3}}\right )^2}{4 e^3}-\frac{b^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^3}{9 e^3}-\frac{3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac{\left (b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{e^3}\\ &=\frac{3 b^2 d n^2 \left (d+\frac{e}{x^{2/3}}\right )^2}{4 e^3}-\frac{b^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^3}{9 e^3}-\frac{3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac{b^2 d^3 n^2 \log ^2\left (d+\frac{e}{x^{2/3}}\right )}{2 e^3}+\frac{1}{6} b n \left (\frac{18 d^2 \left (d+\frac{e}{x^{2/3}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{x^{2/3}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}\\ \end{align*}
Mathematica [C] time = 0.528384, size = 691, normalized size = 2.5 \[ \frac{b n \left (-36 d^3 x^2 \left (b n \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )+\log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )\right )+18 b d^3 n x^2 \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 )+18 b d^3 n x^2 \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 )-36 d^3 x^2 \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 )-36 d^3 x^2 \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 )+36 d^2 x^{4/3} \left (e (a-b n)+b \left (d x^{2/3}+e\right ) \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-18 d e^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-2 b n \left (e \left (6 d^2 x^{4/3}-3 d e x^{2/3}+2 e^2\right )-6 d^3 x^2 \log \left (d+\frac{e}{x^{2/3}}\right )\right )+9 b d n x^{2/3} \left (2 d^2 x^{4/3} \log \left (d+\frac{e}{x^{2/3}}\right )+e \left (e-2 d x^{2/3}\right )\right )\right )-18 e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{36 e^3 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.415, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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 [A] time = 1.0588, size = 402, normalized size = 1.46 \begin{align*} -\frac{1}{6} \, a b e n{\left (\frac{6 \, d^{3} \log \left (d x^{\frac{2}{3}} + e\right )}{e^{4}} - \frac{6 \, d^{3} \log \left (x^{\frac{2}{3}}\right )}{e^{4}} - \frac{6 \, d^{2} x^{\frac{4}{3}} - 3 \, d e x^{\frac{2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} - \frac{1}{36} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (d x^{\frac{2}{3}} + e\right )}{e^{4}} - \frac{6 \, d^{3} \log \left (x^{\frac{2}{3}}\right )}{e^{4}} - \frac{6 \, d^{2} x^{\frac{4}{3}} - 3 \, d e x^{\frac{2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) - \frac{{\left (18 \, d^{3} x^{2} \log \left (d x^{\frac{2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} e x^{\frac{4}{3}} + 15 \, d e^{2} x^{\frac{2}{3}} - 4 \, e^{3} - 6 \,{\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac{2}{3}} + e\right )\right )} n^{2}}{e^{3} x^{2}}\right )} b^{2} - \frac{b^{2} \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac{a b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right )}{x^{2}} - \frac{a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86134, size = 695, normalized size = 2.52 \begin{align*} -\frac{4 \, b^{2} e^{3} n^{2} + 18 \, b^{2} e^{3} \log \left (c\right )^{2} - 12 \, a b e^{3} n + 18 \, a^{2} e^{3} + 18 \,{\left (b^{2} d^{3} n^{2} x^{2} + b^{2} e^{3} n^{2}\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{2} - 12 \,{\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} \log \left (c\right ) - 6 \,{\left (6 \, b^{2} d^{2} e n^{2} x^{\frac{4}{3}} - 3 \, b^{2} d e^{2} n^{2} x^{\frac{2}{3}} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n +{\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x^{2} - 6 \,{\left (b^{2} d^{3} n x^{2} + b^{2} e^{3} n\right )} \log \left (c\right )\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right ) - 3 \,{\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac{2}{3}} - 6 \,{\left (6 \, b^{2} d^{2} e n x \log \left (c\right ) -{\left (11 \, b^{2} d^{2} e n^{2} - 6 \, a b d^{2} e n\right )} x\right )} x^{\frac{1}{3}}}{36 \, e^{3} x^{2}} \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{2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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