Integrand size = 20, antiderivative size = 28 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\frac {\left (-1+x^3\right )^{4/3} \left (-2+4 x^3+3 x^6\right )}{20 x^{10}} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {464, 277, 270} \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=-\frac {\left (x^3-1\right )^{4/3}}{10 x^{10}}+\frac {\left (x^3-1\right )^{4/3}}{5 x^7}+\frac {3 \left (x^3-1\right )^{4/3}}{20 x^4} \]
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Rule 270
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {7}{5} \int \frac {\sqrt [3]{-1+x^3}}{x^8} \, dx \\ & = -\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {\left (-1+x^3\right )^{4/3}}{5 x^7}+\frac {3}{5} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx \\ & = -\frac {\left (-1+x^3\right )^{4/3}}{10 x^{10}}+\frac {\left (-1+x^3\right )^{4/3}}{5 x^7}+\frac {3 \left (-1+x^3\right )^{4/3}}{20 x^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\frac {\left (-1+x^3\right )^{4/3} \left (-2+4 x^3+3 x^6\right )}{20 x^{10}} \]
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Time = 0.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}} \left (3 x^{6}+4 x^{3}-2\right )}{20 x^{10}}\) | \(25\) |
trager | \(\frac {\left (3 x^{9}+x^{6}-6 x^{3}+2\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{20 x^{10}}\) | \(28\) |
gosper | \(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (3 x^{6}+4 x^{3}-2\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{20 x^{10}}\) | \(34\) |
risch | \(\frac {3 x^{12}-2 x^{9}-7 x^{6}+8 x^{3}-2}{20 \left (x^{3}-1\right )^{\frac {2}{3}} x^{10}}\) | \(35\) |
meijerg | \(-\frac {2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{7 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{7}}+\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {9}{14} x^{9}-\frac {3}{14} x^{6}-\frac {1}{7} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{10 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{10}}\) | \(95\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\frac {{\left (3 \, x^{9} + x^{6} - 6 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{20 \, x^{10}} \]
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Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 571, normalized size of antiderivative = 20.39 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=- \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} + 2 \left (\begin {cases} \frac {3 x^{6} \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 ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{3} \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 ) - 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^{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}}} e^{\frac {i \pi }{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 {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{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}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} - \frac {{\left (x^{3} - 1\right )}^{\frac {10}{3}}}{10 \, x^{10}} \]
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\[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\int { \frac {{\left (2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{11}} \,d x } \]
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Time = 5.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{11}} \, dx=\frac {3\,{\left (x^3-1\right )}^{1/3}}{20\,x}+\frac {{\left (x^3-1\right )}^{1/3}}{20\,x^4}-\frac {3\,{\left (x^3-1\right )}^{1/3}}{10\,x^7}+\frac {{\left (x^3-1\right )}^{1/3}}{10\,x^{10}} \]
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