Optimal. Leaf size=39 \[ -\frac{9 a^2 b}{\sqrt [3]{x}}+a^3 \log (x)-\frac{9 a b^2}{2 x^{2/3}}-\frac{b^3}{x} \]
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Rubi [A] time = 0.0231502, antiderivative size = 39, 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} \[ -\frac{9 a^2 b}{\sqrt [3]{x}}+a^3 \log (x)-\frac{9 a b^2}{2 x^{2/3}}-\frac{b^3}{x} \]
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 )^3}{x} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^3}{x^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^3}{x^4}+\frac{3 a b^2}{x^3}+\frac{3 a^2 b}{x^2}+\frac{a^3}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b^3}{x}-\frac{9 a b^2}{2 x^{2/3}}-\frac{9 a^2 b}{\sqrt [3]{x}}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0228175, size = 39, normalized size = 1. \[ -\frac{9 a^2 b}{\sqrt [3]{x}}+a^3 \log (x)-\frac{9 a b^2}{2 x^{2/3}}-\frac{b^3}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{x}}-{\frac{9\,{b}^{2}a}{2}{x}^{-{\frac{2}{3}}}}-9\,{\frac{b{a}^{2}}{\sqrt [3]{x}}}+{a}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99896, size = 45, normalized size = 1.15 \begin{align*} a^{3} \log \left (x\right ) - \frac{9 \, a^{2} b}{x^{\frac{1}{3}}} - \frac{9 \, a b^{2}}{2 \, x^{\frac{2}{3}}} - \frac{b^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50506, size = 101, normalized size = 2.59 \begin{align*} \frac{6 \, a^{3} x \log \left (x^{\frac{1}{3}}\right ) - 18 \, a^{2} b x^{\frac{2}{3}} - 9 \, a b^{2} x^{\frac{1}{3}} - 2 \, b^{3}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.822319, size = 36, normalized size = 0.92 \begin{align*} a^{3} \log{\left (x \right )} - \frac{9 a^{2} b}{\sqrt [3]{x}} - \frac{9 a b^{2}}{2 x^{\frac{2}{3}}} - \frac{b^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22242, size = 50, normalized size = 1.28 \begin{align*} a^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} + 2 \, b^{3}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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