Optimal. Leaf size=48 \[ \frac {4 \left (x^5+x\right )^{3/4} \left (x^8-7 x^7-14 x^6+2 x^4-7 x^3+1\right )}{7 x^6 \left (x^4+1\right )} \]
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Rubi [C] time = 0.57, antiderivative size = 208, normalized size of antiderivative = 4.33, number of steps used = 18, number of rules used = 9, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 1833, 1585, 1478, 449, 1835, 1586, 1844, 364} \begin {gather*} \frac {8 \sqrt [4]{x^4+1} x^5 \, _2F_1\left (\frac {19}{16},\frac {5}{4};\frac {35}{16};-x^4\right )}{19 \sqrt [4]{x^5+x}}-\frac {8 \sqrt [4]{x^4+1} x \, _2F_1\left (\frac {3}{16},\frac {5}{4};\frac {19}{16};-x^4\right )}{\sqrt [4]{x^5+x}}+\frac {4 \sqrt [4]{x^4+1} x^7 \, _2F_1\left (\frac {5}{4},\frac {27}{16};\frac {43}{16};-x^4\right )}{27 \sqrt [4]{x^5+x}}+\frac {4 \sqrt [4]{x^4+1} x^3 \, _2F_1\left (\frac {11}{16},\frac {5}{4};\frac {27}{16};-x^4\right )}{7 \sqrt [4]{x^5+x}}+\frac {8}{7 \sqrt [4]{x^5+x} x}+\frac {4}{7 \sqrt [4]{x^5+x} x^5}-\frac {4 \left (x^4+1\right )}{\sqrt [4]{x^5+x} x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 449
Rule 1478
Rule 1585
Rule 1586
Rule 1833
Rule 1835
Rule 1844
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-3+x^4\right ) \left (1-2 x^3+x^4\right ) \left (1-x^3+x^4\right )}{x^{25/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \left (\frac {9 x^2+6 x^6-3 x^{10}}{x^{21/4} \left (1+x^4\right )^{5/4}}+\frac {-3-5 x^4-6 x^6-x^8+2 x^{10}+x^{12}}{x^{25/4} \left (1+x^4\right )^{5/4}}\right ) \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9 x^2+6 x^6-3 x^{10}}{x^{21/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-3-5 x^4-6 x^6-x^8+2 x^{10}+x^{12}}{x^{25/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x^3+63 x^5+\frac {21 x^7}{2}-21 x^9-\frac {21 x^{11}}{2}}{x^{21/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9+6 x^4-3 x^8}{x^{13/4} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x^2+63 x^4+\frac {21 x^6}{2}-21 x^8-\frac {21 x^{10}}{2}}{x^{17/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {9-3 x^4}{x^{13/4} \sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15 x+63 x^3+\frac {21 x^5}{2}-21 x^7-\frac {21 x^9}{2}}{x^{13/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {15+63 x^2+\frac {21 x^4}{2}-21 x^6-\frac {21 x^8}{2}}{x^{9/4} \left (1+x^4\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-\frac {315 x}{2}+\frac {165 x^3}{4}+\frac {105 x^5}{2}+\frac {105 x^7}{4}}{x^{5/4} \left (1+x^4\right )^{5/4}} \, dx}{105 \sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-\frac {315}{2}+\frac {165 x^2}{4}+\frac {105 x^4}{2}+\frac {105 x^6}{4}}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{105 \sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \left (-\frac {315}{2 \sqrt [4]{x} \left (1+x^4\right )^{5/4}}+\frac {165 x^{7/4}}{4 \left (1+x^4\right )^{5/4}}+\frac {105 x^{15/4}}{2 \left (1+x^4\right )^{5/4}}+\frac {105 x^{23/4}}{4 \left (1+x^4\right )^{5/4}}\right ) \, dx}{105 \sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{23/4}}{\left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}+\frac {\left (11 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\left (1+x^4\right )^{5/4}} \, dx}{7 \sqrt [4]{x+x^5}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{15/4}}{\left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}-\frac {\left (6 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {4}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8}{7 x \sqrt [4]{x+x^5}}-\frac {4 \left (1+x^4\right )}{x^2 \sqrt [4]{x+x^5}}-\frac {8 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {3}{16},\frac {5}{4};\frac {19}{16};-x^4\right )}{\sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {11}{16},\frac {5}{4};\frac {27}{16};-x^4\right )}{7 \sqrt [4]{x+x^5}}+\frac {8 x^5 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {19}{16},\frac {5}{4};\frac {35}{16};-x^4\right )}{19 \sqrt [4]{x+x^5}}+\frac {4 x^7 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {5}{4},\frac {27}{16};\frac {43}{16};-x^4\right )}{27 \sqrt [4]{x+x^5}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 203, normalized size = 4.23 \begin {gather*} \frac {4 \sqrt [4]{x^4+1} \left (129789 \, _2F_1\left (-\frac {21}{16},\frac {5}{4};-\frac {5}{16};-x^4\right )+x^3 \left (778734 x^4 \, _2F_1\left (\frac {7}{16},\frac {5}{4};\frac {23}{16};-x^4\right )+908523 x \, _2F_1\left (-\frac {5}{16},\frac {5}{4};\frac {11}{16};-x^4\right )-908523 \, _2F_1\left (-\frac {9}{16},\frac {5}{4};\frac {7}{16};-x^4\right )+33649 x^9 \, _2F_1\left (\frac {5}{4},\frac {27}{16};\frac {43}{16};-x^4\right )-118503 x^8 \, _2F_1\left (\frac {5}{4},\frac {23}{16};\frac {39}{16};-x^4\right )+95634 x^7 \, _2F_1\left (\frac {19}{16},\frac {5}{4};\frac {35}{16};-x^4\right )-82593 x^5 \, _2F_1\left (\frac {11}{16},\frac {5}{4};\frac {27}{16};-x^4\right )-1817046 x^3 \, _2F_1\left (\frac {3}{16},\frac {5}{4};\frac {19}{16};-x^4\right )\right )\right )}{908523 x^5 \sqrt [4]{x^5+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.63, size = 48, normalized size = 1.00 \begin {gather*} \frac {4 \left (x+x^5\right )^{3/4} \left (1-7 x^3+2 x^4-14 x^6-7 x^7+x^8\right )}{7 x^6 \left (1+x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 43, normalized size = 0.90 \begin {gather*} \frac {4 \, {\left (x^{8} - 7 \, x^{7} - 14 \, x^{6} + 2 \, x^{4} - 7 \, x^{3} + 1\right )} {\left (x^{5} + x\right )}^{\frac {3}{4}}}{7 \, {\left (x^{10} + x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} - 2 \, x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 38, normalized size = 0.79
method | result | size |
gosper | \(\frac {-8 x^{6}-4 x^{7}-4 x^{3}+\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{\left (x^{5}+x \right )^{\frac {1}{4}} x^{5}}\) | \(38\) |
risch | \(\frac {-8 x^{6}-4 x^{7}-4 x^{3}+\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(40\) |
trager | \(\frac {4 \left (x^{5}+x \right )^{\frac {3}{4}} \left (x^{8}-7 x^{7}-14 x^{6}+2 x^{4}-7 x^{3}+1\right )}{7 x^{6} \left (x^{4}+1\right )}\) | \(45\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {21}{16}, \frac {5}{4}\right ], \left [-\frac {5}{16}\right ], -x^{4}\right )}{7 x^{\frac {21}{4}}}+\frac {4 \hypergeom \left (\left [-\frac {5}{16}, \frac {5}{4}\right ], \left [\frac {11}{16}\right ], -x^{4}\right )}{x^{\frac {5}{4}}}-\frac {4 \hypergeom \left (\left [-\frac {9}{16}, \frac {5}{4}\right ], \left [\frac {7}{16}\right ], -x^{4}\right )}{x^{\frac {9}{4}}}-\frac {4 \hypergeom \left (\left [\frac {11}{16}, \frac {5}{4}\right ], \left [\frac {27}{16}\right ], -x^{4}\right ) x^{\frac {11}{4}}}{11}+\frac {24 \hypergeom \left (\left [\frac {7}{16}, \frac {5}{4}\right ], \left [\frac {23}{16}\right ], -x^{4}\right ) x^{\frac {7}{4}}}{7}-8 \hypergeom \left (\left [\frac {3}{16}, \frac {5}{4}\right ], \left [\frac {19}{16}\right ], -x^{4}\right ) x^{\frac {3}{4}}+\frac {4 \hypergeom \left (\left [\frac {5}{4}, \frac {27}{16}\right ], \left [\frac {43}{16}\right ], -x^{4}\right ) x^{\frac {27}{4}}}{27}-\frac {12 \hypergeom \left (\left [\frac {5}{4}, \frac {23}{16}\right ], \left [\frac {39}{16}\right ], -x^{4}\right ) x^{\frac {23}{4}}}{23}+\frac {8 \hypergeom \left (\left [\frac {19}{16}, \frac {5}{4}\right ], \left [\frac {35}{16}\right ], -x^{4}\right ) x^{\frac {19}{4}}}{19}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} - 2 \, x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 53, normalized size = 1.10 \begin {gather*} \frac {4\,{\left (x^5+x\right )}^{3/4}}{7\,x^2}-\frac {8\,{\left (x^5+x\right )}^{3/4}}{x^4+1}-\frac {4\,{\left (x^5+x\right )}^{3/4}}{x^3}+\frac {4\,{\left (x^5+x\right )}^{3/4}}{7\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x^{4} - 3\right ) \left (x^{4} - x^{3} + 1\right ) \left (x^{3} - x^{2} - x - 1\right )}{x^{6} \sqrt [4]{x \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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