Optimal. Leaf size=32 \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]
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Rubi [A] time = 0.0163252, antiderivative size = 32, 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} \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]
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^2} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^2}{x^{8/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^2}{x^6}+\frac{2 a b}{x^5}+\frac{a^2}{x^4}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^2}{5 x^{5/3}}-\frac{3 a b}{2 x^{4/3}}-\frac{a^2}{x}\\ \end{align*}
Mathematica [A] time = 0.0167544, size = 32, normalized size = 1. \[ -\frac{a^2}{x}-\frac{3 a b}{2 x^{4/3}}-\frac{3 b^2}{5 x^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 25, normalized size = 0.8 \begin{align*} -{\frac{3\,{b}^{2}}{5}{x}^{-{\frac{5}{3}}}}-{\frac{3\,ab}{2}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989646, size = 63, normalized size = 1.97 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5}}{5 \, b^{3}} + \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a}{2 \, b^{3}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50915, size = 74, normalized size = 2.31 \begin{align*} -\frac{10 \, a^{2} x + 15 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{10 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.825653, size = 29, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{x} - \frac{3 a b}{2 x^{\frac{4}{3}}} - \frac{3 b^{2}}{5 x^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16178, size = 35, normalized size = 1.09 \begin{align*} -\frac{10 \, a^{2} x^{\frac{2}{3}} + 15 \, a b x^{\frac{1}{3}} + 6 \, b^{2}}{10 \, x^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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