Optimal. Leaf size=38 \[ -\frac{\left (a+b x^3\right )^{4/3} \, _2F_1\left (-\frac{1}{3},1;-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 a x^5} \]
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Rubi [A] time = 0.0168332, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{5}{3},-\frac{1}{3};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 x^5 \sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^6} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{\sqrt [3]{1+\frac{b x^3}{a}}}{x^6} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{5}{3},-\frac{1}{3};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 x^5 \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0091583, size = 51, normalized size = 1.34 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{5}{3},-\frac{1}{3};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 x^5 \sqrt [3]{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}}\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.35982, size = 42, normalized size = 1.11 \begin{align*} \frac{\sqrt [3]{b} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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