Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{-1+x^5} \left (-1+5 x^4+x^5\right )}{5 x^5} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1849, 1599, 12, 267} \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{x^5-1}}{x}-\frac {4 \sqrt [4]{x^5-1}}{5 x^5}+\frac {4}{5} \sqrt [4]{x^5-1} \]
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Rule 12
Rule 267
Rule 1599
Rule 1849
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {1}{10} \int \frac {40 x^3+10 x^8+10 x^9}{x^5 \left (-1+x^5\right )^{3/4}} \, dx \\ & = -\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {1}{10} \int \frac {40+10 x^5+10 x^6}{x^2 \left (-1+x^5\right )^{3/4}} \, dx \\ & = -\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x}+\frac {1}{20} \int \frac {20 x^4}{\left (-1+x^5\right )^{3/4}} \, dx \\ & = -\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x}+\int \frac {x^4}{\left (-1+x^5\right )^{3/4}} \, dx \\ & = \frac {4}{5} \sqrt [4]{-1+x^5}-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x} \\ \end{align*}
Time = 1.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{-1+x^5} \left (-1+5 x^4+x^5\right )}{5 x^5} \]
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Time = 1.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
trager | \(\frac {4 \left (x^{5}-1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}-1\right )}{5 x^{5}}\) | \(23\) |
pseudoelliptic | \(\frac {4 \left (x^{5}-1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}-1\right )}{5 x^{5}}\) | \(23\) |
risch | \(\frac {-\frac {8}{5} x^{5}+\frac {4}{5}+4 x^{9}-4 x^{4}+\frac {4}{5} x^{10}}{\left (x^{5}-1\right )^{\frac {3}{4}} x^{5}}\) | \(33\) |
gosper | \(\frac {4 \left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x^{5}+5 x^{4}-1\right )}{5 x^{5} \left (x^{5}-1\right )^{\frac {3}{4}}}\) | \(38\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [2\right ], x^{5}\right )}{5 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], x^{5}\right )}{4 \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {3 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{5}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+5 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{5 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} \left (-\frac {21 \Gamma \left (\frac {3}{4}\right ) x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], x^{5}\right )}{32}-\frac {3 \left (\frac {1}{3}-3 \ln \left (2\right )+\frac {\pi }{2}+5 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{4}+\frac {\Gamma \left (\frac {3}{4}\right )}{x^{5}}\right )}{5 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{5}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{\operatorname {signum}\left (x^{5}-1\right )^{\frac {3}{4}} x}\) | \(230\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4 \, {\left (x^{5} + 5 \, x^{4} - 1\right )} {\left (x^{5} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 5.81 \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {x^{4} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} + \frac {4 \sqrt [4]{x^{5} - 1}}{5} - \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} - \frac {3 \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {4 \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {35}{4}} \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\int { \frac {{\left (x^{5} + x^{4} - 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
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\[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\int { \frac {{\left (x^{5} + x^{4} - 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
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Time = 5.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx=\frac {4\,{\left (x^5-1\right )}^{5/4}+20\,x^4\,{\left (x^5-1\right )}^{1/4}}{5\,x^5} \]
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