Optimal. Leaf size=28 \[ \frac {\sqrt [3]{x^3+1} \left (-5 x^6-3 x^3+2\right )}{14 x^7} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} \frac {\left (x^3+1\right )^{4/3}}{7 x^7}-\frac {5 \left (x^3+1\right )^{4/3}}{14 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^8} \, dx &=\frac {\left (1+x^3\right )^{4/3}}{7 x^7}+\frac {10}{7} \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx\\ &=\frac {\left (1+x^3\right )^{4/3}}{7 x^7}-\frac {5 \left (1+x^3\right )^{4/3}}{14 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (2-5 x^3\right ) \left (x^3+1\right )^{4/3}}{14 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (2-5 x^3\right ) \left (x^3+1\right )^{4/3}}{14 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 24, normalized size = 0.86 \begin {gather*} -\frac {{\left (5 \, x^{6} + 3 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{14 \, x^{7}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.11 \begin {gather*} -\frac {\left (x^{2}-x +1\right ) \left (1+x \right ) \left (5 x^{3}-2\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{14 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 25, normalized size = 0.89 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} + \frac {{\left (x^{3} + 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 39, normalized size = 1.39 \begin {gather*} -\frac {3\,x^3\,{\left (x^3+1\right )}^{1/3}-2\,{\left (x^3+1\right )}^{1/3}+5\,x^6\,{\left (x^3+1\right )}^{1/3}}{14\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.14, size = 134, normalized size = 4.79 \begin {gather*} \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{3 x \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{7} \Gamma \left (- \frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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