Optimal. Leaf size=33 \[ \frac {\sqrt [3]{x^3-1} \left (12 x^9+4 x^6-9 x^3-7\right )}{70 x^{10}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 271, 264} \begin {gather*} \frac {\left (x^3-1\right )^{4/3}}{10 x^{10}}+\frac {8 \left (x^3-1\right )^{4/3}}{35 x^7}+\frac {6 \left (x^3-1\right )^{4/3}}{35 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{11}} \, dx &=\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8}{5} \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx\\ &=\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8 \left (-1+x^3\right )^{4/3}}{35 x^7}+\frac {24}{35} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx\\ &=\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8 \left (-1+x^3\right )^{4/3}}{35 x^7}+\frac {6 \left (-1+x^3\right )^{4/3}}{35 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{x^3-1} \left (12 x^9+4 x^6-9 x^3-7\right )}{70 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 28, normalized size = 0.85 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (12 x^6+16 x^3+7\right )}{70 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 29, normalized size = 0.88 \begin {gather*} \frac {{\left (12 \, x^{9} + 4 \, x^{6} - 9 \, x^{3} - 7\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{70 \, x^{10}} \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 )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{11}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 1.03 \begin {gather*} \frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (12 x^{6}+16 x^{3}+7\right ) \left (-1+x \right ) \left (x^{2}+x +1\right )}{70 x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 37, normalized size = 1.12 \begin {gather*} \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} - \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} + \frac {{\left (x^{3} - 1\right )}^{\frac {10}{3}}}{10 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 49, normalized size = 1.48 \begin {gather*} \frac {6\,{\left (x^3-1\right )}^{1/3}}{35\,x}+\frac {2\,{\left (x^3-1\right )}^{1/3}}{35\,x^4}-\frac {9\,{\left (x^3-1\right )}^{1/3}}{70\,x^7}-\frac {{\left (x^3-1\right )}^{1/3}}{10\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.82, size = 563, normalized size = 17.06 \begin {gather*} \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{3} \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{9} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \begin {cases} \frac {2 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {2 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {28 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {2 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {2 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \Gamma \left (- \frac {1}{3}\right )} + \frac {28 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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