Integrand size = 30, antiderivative size = 28 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (3-7 x^3+3 x^5\right ) \left (x+x^6\right )^{3/4}}{21 x^6} \]
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Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2077, 2050, 2057, 371} \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \left (x^6+x\right )^{3/4}}{7 x}+\frac {4 \left (x^6+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \]
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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^6}}+\frac {3}{x^3 \sqrt [4]{x+x^6}}-\frac {1}{x \sqrt [4]{x+x^6}}-\frac {2 x^2}{\sqrt [4]{x+x^6}}+\frac {2 x^4}{\sqrt [4]{x+x^6}}\right ) \, dx \\ & = -\left (2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx\right )+2 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-3 \int \frac {1}{x^6 \sqrt [4]{x+x^6}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{x+x^6}} \, dx-\int \frac {1}{x \sqrt [4]{x+x^6}} \, dx \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{x}+\frac {6}{7} \int \frac {1}{x \sqrt [4]{x+x^6}} \, dx+2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx-14 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {8 x^3 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {8 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+12 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {48 x^5 \sqrt [4]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )}{19 \sqrt [4]{x+x^6}}+\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}} \\ & = \frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
Time = 10.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.50 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \sqrt [4]{1+x^5} \left (627 \operatorname {Hypergeometric2F1}\left (-\frac {21}{20},\frac {1}{4},-\frac {1}{20},-x^5\right )+7 x^3 \left (-209 \operatorname {Hypergeometric2F1}\left (-\frac {9}{20},\frac {1}{4},\frac {11}{20},-x^5\right )+627 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{20},\frac {1}{4},\frac {19}{20},-x^5\right )-114 x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{20},\frac {31}{20},-x^5\right )+66 x^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {19}{20},\frac {39}{20},-x^5\right )\right )\right )}{4389 x^5 \sqrt [4]{x+x^6}} \]
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Time = 0.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {4 \left (3 x^{5}-7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(25\) |
pseudoelliptic | \(\frac {4 \left (3 x^{5}-7 x^{3}+3\right ) \left (x^{6}+x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(25\) |
risch | \(\frac {\frac {4}{7} x^{10}+\frac {8}{7} x^{5}+\frac {4}{7}-\frac {4}{3} x^{8}-\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{5}+1\right )\right )}^{\frac {1}{4}}}\) | \(37\) |
gosper | \(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (3 x^{5}-7 x^{3}+3\right )}{21 x^{5} \left (x^{6}+x \right )^{\frac {1}{4}}}\) | \(44\) |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {21}{20}, \frac {1}{4}\right ], \left [-\frac {1}{20}\right ], -x^{5}\right )}{7 x^{\frac {21}{4}}}+\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{20}, \frac {1}{4}\right ], \left [\frac {19}{20}\right ], -x^{5}\right )}{x^{\frac {1}{4}}}+\frac {8 x^{\frac {19}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {19}{20}\right ], \left [\frac {39}{20}\right ], -x^{5}\right )}{19}-\frac {8 x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {11}{20}\right ], \left [\frac {31}{20}\right ], -x^{5}\right )}{11}-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {9}{20}, \frac {1}{4}\right ], \left [\frac {11}{20}\right ], -x^{5}\right )}{3 x^{\frac {9}{4}}}\) | \(82\) |
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} - 7 \, x^{3} + 3\right )}}{21 \, x^{6}} \]
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\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int \frac {\left (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]
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\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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\[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Time = 5.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx=\frac {12\,{\left (x^6+x\right )}^{3/4}-28\,x^3\,{\left (x^6+x\right )}^{3/4}+12\,x^5\,{\left (x^6+x\right )}^{3/4}}{21\,x^6} \]
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