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 = 0.02, 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 364
Rule 365
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*}
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Mathematica [A] time = 0.01, 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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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.17, size = 42, normalized size = 1.11 \[ \frac {\sqrt [3]{b} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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