Optimal. Leaf size=28 \[ \frac {\left (x^3-1\right )^{2/3} \left (7 x^6-2 x^3-5\right )}{20 x^8} \]
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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 )^{5/3}}{4 x^8}+\frac {7 \left (x^3-1\right )^{5/3}}{20 x^5} \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 )^{2/3} \left (2+x^3\right )}{x^9} \, dx &=\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {7}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {7 \left (-1+x^3\right )^{5/3}}{20 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (x^3-1\right )^{5/3} \left (7 x^3+5\right )}{20 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-5-2 x^3+7 x^6\right )}{20 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 24, normalized size = 0.86 \begin {gather*} \frac {{\left (7 \, x^{6} - 2 \, x^{3} - 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \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} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 25, normalized size = 0.89
method | result | size |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (7 x^{6}-2 x^{3}-5\right )}{20 x^{8}}\) | \(25\) |
gosper | \(\frac {\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (7 x^{3}+5\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(29\) |
risch | \(\frac {7 x^{9}-9 x^{6}-3 x^{3}+5}{20 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(30\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{5}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{8}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 24, normalized size = 0.86 \begin {gather*} -\frac {{\left (x^3-1\right )}^{2/3}\,\left (-7\,x^6+2\,x^3+5\right )}{20\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.56, size = 420, normalized size = 15.00 \begin {gather*} \begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases} + 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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