Integrand size = 23, antiderivative size = 16 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (x+x^5\right )^{7/4}}{7 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2077, 2050, 2057, 371} \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (x^5+x\right )^{3/4}}{7 x^6}+\frac {4 \left (x^5+x\right )^{3/4}}{7 x^2} \]
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Rule 371
Rule 2050
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3}{x^6 \sqrt [4]{x+x^5}}-\frac {2}{x^2 \sqrt [4]{x+x^5}}+\frac {x^2}{\sqrt [4]{x+x^5}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx\right )-3 \int \frac {1}{x^6 \sqrt [4]{x+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx \\ & = \frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {8 \left (x+x^5\right )^{3/4}}{5 x^2}+\frac {9}{7} \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx-\frac {14}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{11 \sqrt [4]{x+x^5}}+\frac {9}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}} \\ & = \frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}-\frac {36 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{55 \sqrt [4]{x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}} \\ & = \frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2} \\ \end{align*}
Time = 10.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6} \]
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Time = 0.96 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
trager | \(\frac {4 \left (x^{4}+1\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{7 x^{6}}\) | \(18\) |
gosper | \(\frac {4 \left (x^{4}+1\right )^{2}}{7 x^{5} \left (x^{5}+x \right )^{\frac {1}{4}}}\) | \(20\) |
pseudoelliptic | \(\frac {4 \left (x^{4}+1\right ) {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}}{7 x^{6}}\) | \(20\) |
risch | \(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\) | \(25\) |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], -x^{4}\right )}{7 x^{\frac {21}{4}}}+\frac {8 \operatorname {hypergeom}\left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], -x^{4}\right )}{5 x^{\frac {5}{4}}}+\frac {4 x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], -x^{4}\right )}{11}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (x^{4} + 1\right )}}{7 \, x^{6}} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (x^{4} - 3\right ) \left (x^{4} + 1\right )}{x^{6} \sqrt [4]{x \left (x^{4} + 1\right )}}\, dx \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Time = 5.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx=\frac {4\,{\left (x^5+x\right )}^{3/4}+4\,x^4\,{\left (x^5+x\right )}^{3/4}}{7\,x^6} \]
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