Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (1+x^8\right )^{3/2}}{6 x^6} \]
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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} \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (x^8+1\right )^{3/2}}{6 x^6} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^8\right )^{3/2}}{6 x^6} \\ \end{align*}
Time = 2.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {\left (1+x^8\right )^{3/2}}{6 x^6} \]
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Time = 1.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(13\) |
trager | \(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(13\) |
pseudoelliptic | \(\frac {\left (x^{8}+1\right ) \sqrt {\frac {x^{8}+1}{x^{2}}}}{6 x^{5}}\) | \(22\) |
risch | \(\frac {x^{16}+2 x^{8}+1}{6 x^{6} \sqrt {x^{8}+1}}\) | \(23\) |
meijerg | \(\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{8}\right )}{6 x^{6}}+\frac {x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{8}\right )}{2}\) | \(34\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.12 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} - \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x^{6} \Gamma \left (\frac {1}{4}\right )} \]
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none
Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {1}{6} \, \sqrt {x^{8} + 1} x^{2} + \frac {\sqrt {\frac {1}{x^{8}} + 1}}{6 \, x^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx=\frac {{\left (x^8+1\right )}^{3/2}}{6\,x^6} \]
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