Optimal. Leaf size=38 \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{3}{a \left (a+b \sqrt [3]{x}\right )} \]
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Rubi [A] time = 0.0220806, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{3}{a \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2 x} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{a \left (a+b \sqrt [3]{x}\right )}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^2}+\frac{\log (x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0224447, size = 33, normalized size = 0.87 \[ \frac{\frac{3 a}{a+b \sqrt [3]{x}}-3 \log \left (a+b \sqrt [3]{x}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 0.9 \begin{align*} 3\,{\frac{1}{a \left ( a+b\sqrt [3]{x} \right ) }}-3\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{2}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958126, size = 46, normalized size = 1.21 \begin{align*} \frac{3}{a b x^{\frac{1}{3}} + a^{2}} - \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.56633, size = 166, normalized size = 4.37 \begin{align*} \frac{3 \,{\left (a b^{2} x^{\frac{2}{3}} - a^{2} b x^{\frac{1}{3}} + a^{3} -{\left (b^{3} x + a^{3}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (b^{3} x + a^{3}\right )} \log \left (x^{\frac{1}{3}}\right )\right )}}{a^{2} b^{3} x + a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19275, size = 160, normalized size = 4.21 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{2 b^{2} x^{\frac{2}{3}}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\\frac{a x^{\frac{2}{3}} \log{\left (x \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} - \frac{3 a x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} + \frac{3 a x^{\frac{2}{3}}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} + \frac{b x \log{\left (x \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} - \frac{3 b x \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{3} x^{\frac{2}{3}} + a^{2} b x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23448, size = 49, normalized size = 1.29 \begin{align*} -\frac{3 \, \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a^{2}} + \frac{3}{{\left (b x^{\frac{1}{3}} + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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