Integrand size = 18, antiderivative size = 14 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5}}{x} \]
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Time = 0.00 (sec) , antiderivative size = 14, 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} \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{x^5+1}}{x} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {4 \sqrt [4]{1+x^5}}{x} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{1+x^5}}{x} \]
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Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
trager | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
pseudoelliptic | \(\frac {4 \left (x^{5}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right )}{x \left (x^{5}+1\right )^{\frac {3}{4}}}\) | \(32\) |
meijerg | \(\frac {x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], -x^{5}\right )}{4}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}\) | \(34\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{5} + 1\right )}^{\frac {1}{4}}}{x} \]
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Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.50 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {x^{4} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} - \frac {4 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\int { \frac {x^{5} - 4}{{\left (x^{5} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
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Time = 4.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {-4+x^5}{x^2 \left (1+x^5\right )^{3/4}} \, dx=\frac {4\,{\left (x^5+1\right )}^{1/4}}{x} \]
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