Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\frac {4 \left (-1+x^5\right )^{7/4}}{7 x^7} \]
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Time = 0.01 (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} \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\frac {4 \left (x^5-1\right )^{7/4}}{7 x^7} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-1+x^5\right )^{7/4}}{7 x^7} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\frac {4 \left (-1+x^5\right )^{7/4}}{7 x^7} \]
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Time = 0.90 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {4 \left (x^{5}-1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
pseudoelliptic | \(\frac {4 \left (x^{5}-1\right )^{\frac {7}{4}}}{7 x^{7}}\) | \(13\) |
risch | \(\frac {\frac {4}{7} x^{10}-\frac {8}{7} x^{5}+\frac {4}{7}}{x^{7} \left (x^{5}-1\right )^{\frac {1}{4}}}\) | \(23\) |
gosper | \(\frac {4 \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x -1\right ) \left (x^{5}-1\right )^{\frac {3}{4}}}{7 x^{7}}\) | \(28\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], x^{5}\right )}{2 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{2}}-\frac {4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], x^{5}\right )}{7 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{7}}\) | \(66\) |
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\frac {4 \, {\left (x^{5} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 80, normalized size of antiderivative = 5.00 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=- \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {4 e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\frac {4 \, {\left (x^{5} - 1\right )} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}}}{7 \, x^{7}} \]
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\[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=\int { \frac {{\left (x^{5} + 4\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}{x^{8}} \,d x } \]
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Time = 5.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-1+x^5\right )^{3/4} \left (4+x^5\right )}{x^8} \, dx=-\frac {4\,{\left (x^5-1\right )}^{3/4}-4\,x^5\,{\left (x^5-1\right )}^{3/4}}{7\,x^7} \]
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