Optimal. Leaf size=33 \[ \frac{3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]
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Rubi [A] time = 0.0214783, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ \frac{3 b}{a^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^2 \sqrt [3]{x}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{b}{a (b+a x)^2}+\frac{1}{a (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b}{a^2 \left (b+a \sqrt [3]{x}\right )}+\frac{3 \log \left (b+a \sqrt [3]{x}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0293208, size = 33, normalized size = 1. \[ \frac{-\frac{3 a}{a+\frac{b}{\sqrt [3]{x}}}+3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 30, normalized size = 0.9 \begin{align*} 3\,{\frac{b}{{a}^{2} \left ( b+a\sqrt [3]{x} \right ) }}+3\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974238, size = 46, normalized size = 1.39 \begin{align*} -\frac{3}{a^{2} + \frac{a b}{x^{\frac{1}{3}}}} + \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\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 [A] time = 1.48947, size = 127, normalized size = 3.85 \begin{align*} \frac{3 \,{\left (a^{2} b x^{\frac{2}{3}} - a b^{2} x^{\frac{1}{3}} + b^{3} +{\left (a^{3} x + b^{3}\right )} \log \left (a x^{\frac{1}{3}} + b\right )\right )}}{a^{5} x + a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39864, size = 99, normalized size = 3. \begin{align*} \begin{cases} \frac{3 a x \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} + \frac{3 b x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} + \frac{3 b x^{\frac{2}{3}}}{a^{3} x + a^{2} b x^{\frac{2}{3}}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{2}{3}}}{2 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17113, size = 41, normalized size = 1.24 \begin{align*} \frac{3 \, \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{2}} + \frac{3 \, b}{{\left (a x^{\frac{1}{3}} + b\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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