Optimal. Leaf size=45 \[ -\frac{9 a^2 b}{4 x^{4/3}}-\frac{a^3}{x}-\frac{9 a b^2}{5 x^{5/3}}-\frac{b^3}{2 x^2} \]
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Rubi [A] time = 0.0220165, antiderivative size = 45, 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} \[ -\frac{9 a^2 b}{4 x^{4/3}}-\frac{a^3}{x}-\frac{9 a b^2}{5 x^{5/3}}-\frac{b^3}{2 x^2} \]
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^2} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^3}{x^3} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^3}{x^7}+\frac{3 a b^2}{x^6}+\frac{3 a^2 b}{x^5}+\frac{a^3}{x^4}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b^3}{2 x^2}-\frac{9 a b^2}{5 x^{5/3}}-\frac{9 a^2 b}{4 x^{4/3}}-\frac{a^3}{x}\\ \end{align*}
Mathematica [A] time = 0.0159343, size = 41, normalized size = 0.91 \[ -\frac{45 a^2 b x^{2/3}+20 a^3 x+36 a b^2 \sqrt [3]{x}+10 b^3}{20 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{2\,{x}^{2}}}-{\frac{9\,{b}^{2}a}{5}{x}^{-{\frac{5}{3}}}}-{\frac{9\,b{a}^{2}}{4}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{3}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949698, size = 63, normalized size = 1.4 \begin{align*} -\frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6}}{2 \, b^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a}{5 \, b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{2}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47181, size = 93, normalized size = 2.07 \begin{align*} -\frac{20 \, a^{3} x + 45 \, a^{2} b x^{\frac{2}{3}} + 36 \, a b^{2} x^{\frac{1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.38975, size = 41, normalized size = 0.91 \begin{align*} - \frac{a^{3}}{x} - \frac{9 a^{2} b}{4 x^{\frac{4}{3}}} - \frac{9 a b^{2}}{5 x^{\frac{5}{3}}} - \frac{b^{3}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17475, size = 47, normalized size = 1.04 \begin{align*} -\frac{20 \, a^{3} x + 45 \, a^{2} b x^{\frac{2}{3}} + 36 \, a b^{2} x^{\frac{1}{3}} + 10 \, b^{3}}{20 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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