Optimal. Leaf size=104 \[ \frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{x+x^6}}{-x^2+\sqrt {x+x^6}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x+x^6}}{\sqrt {2}}}{x \sqrt [4]{x+x^6}}\right ) \]
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Rubi [F]
time = 1.59, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^{13/4} \sqrt [4]{1+x^5} \left (1+x^3+x^5\right )} \, dx}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {\left (1-x^{12}+x^{20}\right ) \left (-3+2 x^{20}\right )}{x^{10} \sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \left (-\frac {3}{x^{10} \sqrt [4]{1+x^{20}}}-\frac {4 x^2}{\sqrt [4]{1+x^{20}}}+\frac {2 x^{10}}{\sqrt [4]{1+x^{20}}}+\frac {2 x^2 \left (5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \sqrt [4]{1+x^5} \, _2F_1\left (-\frac {9}{20},\frac {1}{4};\frac {11}{20};-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}-\frac {16 x \sqrt [4]{1+x^5} \, _2F_1\left (\frac {3}{20},\frac {1}{4};\frac {23}{20};-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \left (\frac {5 x^2}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}+\frac {2 x^{14}}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \sqrt [4]{1+x^5} \, _2F_1\left (-\frac {9}{20},\frac {1}{4};\frac {11}{20};-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}-\frac {16 x \sqrt [4]{1+x^5} \, _2F_1\left (\frac {3}{20},\frac {1}{4};\frac {23}{20};-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (40 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ \end {align*}
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Mathematica [F]
time = 30.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 22.84, size = 213, normalized size = 2.05
method | result | size |
trager | \(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+x}\, x +2 \left (x^{6}+x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{5}+x^{3}+1}\right )-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+x}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )\) | \(213\) |
risch | \(\frac {\frac {4 x^{5}}{3}+\frac {4}{3}}{x^{2} \left (x \left (x^{5}+1\right )\right )^{\frac {1}{4}}}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \left (x^{6}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}+x}\, x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{6}+x \right )^{\frac {3}{4}}}{x^{5}+x^{3}+1}\right )-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}+x}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )\) | \(219\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 682 vs.
\(2 (84) = 168\).
time = 47.43, size = 682, normalized size = 6.56 \begin {gather*} \frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{10} + 2 \, x^{8} + x^{6} + 2 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (x^{5} - 3 \, x^{3} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{7} - x^{5} + 3 \, x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{4} + x\right )} \sqrt {x^{6} + x} + {\left (16 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} x^{3} + 2 \, \sqrt {2} {\left (x^{6} - 3 \, x^{4} + x\right )} \sqrt {x^{6} + x} + \sqrt {2} {\left (x^{10} - 8 \, x^{8} - x^{6} + 2 \, x^{5} - 8 \, x^{3} + 1\right )} + 4 \, {\left (x^{7} + x^{5} + x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} + x^{3} + 1}} + 1}{x^{10} - 14 \, x^{8} + x^{6} + 2 \, x^{5} - 14 \, x^{3} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{10} + 2 \, x^{8} + x^{6} + 2 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (x^{5} - 3 \, x^{3} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{7} - x^{5} + 3 \, x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{4} + x\right )} \sqrt {x^{6} + x} + {\left (16 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} x^{3} - 2 \, \sqrt {2} {\left (x^{6} - 3 \, x^{4} + x\right )} \sqrt {x^{6} + x} - \sqrt {2} {\left (x^{10} - 8 \, x^{8} - x^{6} + 2 \, x^{5} - 8 \, x^{3} + 1\right )} + 4 \, {\left (x^{7} + x^{5} + x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} + x^{3} + 1}} + 1}{x^{10} - 14 \, x^{8} + x^{6} + 2 \, x^{5} - 14 \, x^{3} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1\right )}}{x^{5} + x^{3} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1\right )}}{x^{5} + x^{3} + 1}\right ) + 8 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{6 \, x^{3}} \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 (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} + x^{3} + 1\right )}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^5-3\right )\,\left (x^5-x^3+1\right )}{x^3\,{\left (x^6+x\right )}^{1/4}\,\left (x^5+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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