Optimal. Leaf size=28 \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]
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Rubi [A] time = 0.0154615, antiderivative size = 28, 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} \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2}{x} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^2}{x^{5/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^2}{x^3}+\frac{2 a b}{x^2}+\frac{a^2}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^2}{2 x^{2/3}}-\frac{6 a b}{\sqrt [3]{x}}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0183545, size = 28, normalized size = 1. \[ a^2 \log (x)-\frac{6 a b}{\sqrt [3]{x}}-\frac{3 b^2}{2 x^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 23, normalized size = 0.8 \begin{align*} -{\frac{3\,{b}^{2}}{2}{x}^{-{\frac{2}{3}}}}-6\,{\frac{ab}{\sqrt [3]{x}}}+{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988304, size = 30, normalized size = 1.07 \begin{align*} a^{2} \log \left (x\right ) - \frac{6 \, a b}{x^{\frac{1}{3}}} - \frac{3 \, b^{2}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44463, size = 81, normalized size = 2.89 \begin{align*} \frac{3 \,{\left (2 \, a^{2} x \log \left (x^{\frac{1}{3}}\right ) - 4 \, a b x^{\frac{2}{3}} - b^{2} x^{\frac{1}{3}}\right )}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.477558, size = 27, normalized size = 0.96 \begin{align*} a^{2} \log{\left (x \right )} - \frac{6 a b}{\sqrt [3]{x}} - \frac{3 b^{2}}{2 x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19777, size = 32, normalized size = 1.14 \begin{align*} a^{2} \log \left ({\left | x \right |}\right ) - \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + b^{2}\right )}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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