Integrand size = 18, antiderivative size = 28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5-2 x^3+7 x^6\right )}{20 x^8} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {464, 270} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (x^3-1\right )^{5/3}}{4 x^8}+\frac {7 \left (x^3-1\right )^{5/3}}{20 x^5} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5-2 x^3+7 x^6\right )}{20 x^8} \]
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Time = 0.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {\left (7 x^{3}+5\right ) \left (x^{3}-1\right )^{\frac {5}{3}}}{20 x^{8}}\) | \(20\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (7 x^{6}-2 x^{3}-5\right )}{20 x^{8}}\) | \(25\) |
gosper | \(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (7 x^{3}+5\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(29\) |
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 | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{5}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{8}}\) | \(78\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {{\left (7 \, x^{6} - 2 \, x^{3} - 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 428, normalized size of antiderivative = 15.29 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\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 {3 x^{6} \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 ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \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 ) - 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^{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}} e^{\frac {2 i \pi }{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 {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{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}} \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}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Time = 5.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx=-\frac {{\left (x^3-1\right )}^{2/3}\,\left (-7\,x^6+2\,x^3+5\right )}{20\,x^8} \]
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