Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (x^3+1\right )^{5/2}}{5 x^5} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \]
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Time = 2.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {2 \left (x^{3}+1\right )^{\frac {5}{2}}}{5 x^{5}}\) | \(13\) |
trager | \(\frac {2 \left (x^{6}+2 x^{3}+1\right ) \sqrt {x^{3}+1}}{5 x^{5}}\) | \(23\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )^{\frac {3}{2}}}{5 x^{5}}\) | \(24\) |
risch | \(\frac {\frac {2}{5} x^{9}+\frac {6}{5} x^{6}+\frac {6}{5} x^{3}+\frac {2}{5}}{x^{5} \sqrt {x^{3}+1}}\) | \(28\) |
meijerg | \(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {5}{3}, -\frac {3}{2}\right ], \left [-\frac {2}{3}\right ], -x^{3}\right )}{5 x^{5}}-\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{2}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}\) | \(34\) |
default | \(\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}\) | \(36\) |
elliptic | \(\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}\) | \(36\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{3} + 1}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 6.56 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{5 \, x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2}{5} \, \sqrt {x^{3} + 1} x + \frac {2}{5} \, \sqrt {\frac {1}{x} + \frac {1}{x^{4}}} {\left (\frac {1}{x^{3}} + 2\right )} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx=\frac {2\,{\left (x^3+1\right )}^{5/2}}{5\,x^5} \]
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