Optimal. Leaf size=77 \[ -\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 b d n}{e \sqrt [3]{x}}+\frac{2 b n}{3 x} \]
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Rubi [A] time = 0.0509336, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2455, 263, 341, 325, 205} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 b d n}{e \sqrt [3]{x}}+\frac{2 b n}{3 x} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 263
Rule 341
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x^2} \, dx &=-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-\frac{1}{3} (2 b e n) \int \frac{1}{\left (d+\frac{e}{x^{2/3}}\right ) x^{8/3}} \, dx\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-\frac{1}{3} (2 b e n) \int \frac{1}{\left (e+d x^{2/3}\right ) x^2} \, dx\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-(2 b e n) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n}{3 x}-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}+(2 b d n) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n}{3 x}-\frac{2 b d n}{e \sqrt [3]{x}}-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}-\frac{\left (2 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{2 b n}{3 x}-\frac{2 b d n}{e \sqrt [3]{x}}-\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0489569, size = 80, normalized size = 1.04 \[ -\frac{a}{x}-\frac{b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{x}+\frac{2 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{2 b d n}{e \sqrt [3]{x}}+\frac{2 b n}{3 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) }\, 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 [A] time = 1.87208, size = 554, normalized size = 7.19 \begin{align*} \left [\frac{3 \, b d n x \sqrt{-\frac{d}{e}} \log \left (\frac{d^{3} x^{2} + 2 \, d e^{2} x \sqrt{-\frac{d}{e}} - e^{3} - 2 \,{\left (d^{2} e x \sqrt{-\frac{d}{e}} - d e^{2}\right )} x^{\frac{2}{3}} - 2 \,{\left (d^{2} e x + e^{3} \sqrt{-\frac{d}{e}}\right )} x^{\frac{1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 3 \, b e n \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right ) - 6 \, b d n x^{\frac{2}{3}} + 2 \, b e n - 3 \, b e \log \left (c\right ) - 3 \, a e}{3 \, e x}, -\frac{6 \, b d n x \sqrt{\frac{d}{e}} \arctan \left (x^{\frac{1}{3}} \sqrt{\frac{d}{e}}\right ) + 3 \, b e n \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right ) + 6 \, b d n x^{\frac{2}{3}} - 2 \, b e n + 3 \, b e \log \left (c\right ) + 3 \, a e}{3 \, e 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 [A] time = 1.34372, size = 99, normalized size = 1.29 \begin{align*} -\frac{1}{3} \,{\left (2 \,{\left (3 \, d^{\frac{3}{2}} \arctan \left (\sqrt{d} x^{\frac{1}{3}} e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{5}{2}\right )} + \frac{{\left (3 \, d x^{\frac{2}{3}} - e\right )} e^{\left (-2\right )}}{x}\right )} e + \frac{3 \, \log \left (d + \frac{e}{x^{\frac{2}{3}}}\right )}{x}\right )} b n - \frac{b \log \left (c\right )}{x} - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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