Optimal. Leaf size=34 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]
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Rubi [A] time = 0.0159659, antiderivative size = 34, 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} \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]
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^4} \, dx &=\int \frac{\left (b+a \sqrt [3]{x}\right )^2}{x^{14/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^2}{x^{12}}+\frac{2 a b}{x^{11}}+\frac{a^2}{x^{10}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^2}{11 x^{11/3}}-\frac{3 a b}{5 x^{10/3}}-\frac{a^2}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0223455, size = 34, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{5 x^{10/3}}-\frac{3 b^2}{11 x^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 25, normalized size = 0.7 \begin{align*} -{\frac{3\,{b}^{2}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{3\,ab}{5}{x}^{-{\frac{10}{3}}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02447, size = 201, normalized size = 5.91 \begin{align*} -\frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{11}}{11 \, b^{9}} + \frac{12 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{10} a}{5 \, b^{9}} - \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9} a^{2}}{3 \, b^{9}} + \frac{21 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a^{3}}{b^{9}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{4}}{b^{9}} + \frac{28 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{5}}{b^{9}} - \frac{84 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{6}}{5 \, b^{9}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{7}}{b^{9}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{8}}{b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42835, size = 77, normalized size = 2.26 \begin{align*} -\frac{55 \, a^{2} x + 99 \, a b x^{\frac{2}{3}} + 45 \, b^{2} x^{\frac{1}{3}}}{165 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.95926, size = 32, normalized size = 0.94 \begin{align*} - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{5 x^{\frac{10}{3}}} - \frac{3 b^{2}}{11 x^{\frac{11}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20799, size = 35, normalized size = 1.03 \begin{align*} -\frac{55 \, a^{2} x^{\frac{2}{3}} + 99 \, a b x^{\frac{1}{3}} + 45 \, b^{2}}{165 \, x^{\frac{11}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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