Optimal. Leaf size=135 \[ \frac {4 \sqrt [4]{x^5-2 x^4+1}}{x}-2^{3/4} \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5-2 x^4+1}}{\sqrt {2} x^2-\sqrt {x^5-2 x^4+1}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^5-2 x^4+1}}{2 x^2+\sqrt {2} \sqrt {x^5-2 x^4+1}}\right ) \]
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Rubi [F] time = 1.36, 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 (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx &=\int \left (\frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x}-\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x^2}+\frac {\left (1-2 x+3 x^2+x^3\right ) \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}\right ) \, dx\\ &=-\left (4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx\right )+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \frac {\left (1-2 x+3 x^2+x^3\right ) \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx\\ &=-\left (4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx\right )+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \left (\frac {\sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}-\frac {2 x \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}+\frac {3 x^2 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}+\frac {x^3 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {x \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx\right )+3 \int \frac {x^2 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx-4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx+\int \frac {x^3 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.95, size = 135, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 91.55, size = 809, normalized size = 5.99 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {2 \, x^{10} + 4 \, x^{5} + 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 8 \, x^{5} + x\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {2} {\left (x^{7} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x^{5} + 2^{\frac {3}{4}} {\left (x^{10} - 20 \, x^{9} + 32 \, x^{8} + 2 \, x^{5} - 20 \, x^{4} + 1\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{7} - 8 \, x^{6} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} + 8 \, {\left (x^{8} + x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{5} + 1\right )}}{x^{5} + 1}} + 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{8} - 8 \, x^{7} + 3 \, x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} + 2}{2 \, {\left (x^{10} - 32 \, x^{9} + 64 \, x^{8} + 2 \, x^{5} - 32 \, x^{4} + 1\right )}}\right ) - 4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {2 \, x^{10} + 4 \, x^{5} - 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 8 \, x^{5} + x\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {2} {\left (x^{7} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x^{5} - 2^{\frac {3}{4}} {\left (x^{10} - 20 \, x^{9} + 32 \, x^{8} + 2 \, x^{5} - 20 \, x^{4} + 1\right )} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{7} - 8 \, x^{6} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} + 8 \, {\left (x^{8} + x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{5} + 1\right )}}{x^{5} + 1}} - 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{8} - 8 \, x^{7} + 3 \, x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} + 2}{2 \, {\left (x^{10} - 32 \, x^{9} + 64 \, x^{8} + 2 \, x^{5} - 32 \, x^{4} + 1\right )}}\right ) + 2^{\frac {3}{4}} x \log \left (\frac {2 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{5} + 1\right )}\right )}}{x^{5} + 1}\right ) - 2^{\frac {3}{4}} x \log \left (-\frac {2 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{5} + 1\right )}\right )}}{x^{5} + 1}\right ) - 16 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x} \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^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 66.00, size = 332, normalized size = 2.46
method | result | size |
trager | \(\frac {4 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x}+\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{5}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) \sqrt {x^{5}-2 x^{4}+1}\, x^{2}+4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \sqrt {x^{5}-2 x^{4}+1}\, x^{2}+4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )\) | \(332\) |
risch | \(\frac {4 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{15}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{14}+2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x^{11}+20 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{13}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x^{10}-16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{12}+8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x^{9}+3 x^{10} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{9}+4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x^{6}+20 x^{8} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x^{5}-4 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{7}+8 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{6}+3 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {3}{4}} x^{3}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} x -4 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}}{\left (-1+x \right )^{2} \left (x^{4}-x^{3}-x^{2}-x -1\right )^{2} \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{15}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{14}+2 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{11}+20 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{13}-8 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{10}-16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{12}+8 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{9}+3 x^{10} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{9}+4 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+20 x^{8} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-8 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{5}+4 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{7}-8 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{6}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {3}{4}} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{4}+2 \left (x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x +4 \sqrt {x^{15}-6 x^{14}+12 x^{13}-8 x^{12}+3 x^{10}-12 x^{9}+12 x^{8}+3 x^{5}-6 x^{4}+1}\, x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}}{\left (-1+x \right )^{2} \left (x^{4}-x^{3}-x^{2}-x -1\right )^{2} \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )\right ) \left (\left (x^{5}-2 x^{4}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}}}\) | \(1650\) |
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^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}}\,{d x} \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 (x^5-4\right )\,{\left (x^5-2\,x^4+1\right )}^{1/4}}{x^2\,\left (x^5+1\right )} \,d x \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 {\sqrt [4]{\left (x - 1\right ) \left (x^{4} - x^{3} - x^{2} - x - 1\right )} \left (x^{5} - 4\right )}{x^{2} \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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