Optimal. Leaf size=74 \[ \frac {\sqrt {-x^4+x^2+2 x+1} \left (2 x^4+3 x^3+x^2-4 x-2\right )}{6 (x+1)^3}-\tan ^{-1}\left (\frac {\sqrt {-x^4+x^2+2 x+1}}{x^2+x+1}\right ) \]
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Rubi [F] time = 1.11, 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+x^2\right ) \left (2 x+x^2\right ) \sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x+x^2\right ) \left (2 x+x^2\right ) \sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx &=\int \frac {x (2+x) \left (1+x+x^2\right ) \sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx\\ &=\int \left (\sqrt {1+2 x+x^2-x^4}+\frac {\sqrt {1+2 x+x^2-x^4}}{-1-x}-\frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^4}+\frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^3}\right ) \, dx\\ &=\int \sqrt {1+2 x+x^2-x^4} \, dx+\int \frac {\sqrt {1+2 x+x^2-x^4}}{-1-x} \, dx-\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx+\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^3} \, dx\\ &=-\left (16 \operatorname {Subst}\left (\int \frac {\sqrt {\frac {-240-128 x^2+256 x^4}{(2-4 x)^4}}}{(2-4 x)^2} \, dx,x,\frac {1}{2}+\frac {1}{x}\right )\right )+\int \frac {\sqrt {1+2 x+x^2-x^4}}{-1-x} \, dx-\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx+\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^3} \, dx\\ &=-\frac {\left (256 \sqrt {1+2 x+x^2-x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-240-128 x^2+256 x^4}}{(2-4 x)^4} \, dx,x,\frac {1}{2}+\frac {1}{x}\right )}{\sqrt {-240-128 \left (\frac {1}{2}+\frac {1}{x}\right )^2+256 \left (\frac {1}{2}+\frac {1}{x}\right )^4} x^2}+\int \frac {\sqrt {1+2 x+x^2-x^4}}{-1-x} \, dx-\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^4} \, dx+\int \frac {\sqrt {1+2 x+x^2-x^4}}{(1+x)^3} \, dx\\ \end {align*}
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Mathematica [C] time = 6.17, size = 1735, normalized size = 23.45
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.45, size = 74, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+2 x+x^2-x^4} \left (-2-4 x+x^2+3 x^3+2 x^4\right )}{6 (1+x)^3}-\tan ^{-1}\left (\frac {\sqrt {1+2 x+x^2-x^4}}{1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 103, normalized size = 1.39 \begin {gather*} -\frac {3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + x^{2} + 2 \, x + 1} x^{2}}{x^{4} - x^{2} - 2 \, x - 1}\right ) - {\left (2 \, x^{4} + 3 \, x^{3} + x^{2} - 4 \, x - 2\right )} \sqrt {-x^{4} + x^{2} + 2 \, x + 1}}{6 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\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 {\sqrt {-x^{4} + x^{2} + 2 \, x + 1} {\left (x^{2} + 2 \, x\right )} {\left (x^{2} + x + 1\right )}}{{\left (x + 1\right )}^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.21, size = 83, normalized size = 1.12
method | result | size |
trager | \(\frac {\sqrt {-x^{4}+x^{2}+2 x +1}\, \left (2 x^{4}+3 x^{3}+x^{2}-4 x -2\right )}{6 \left (1+x \right )^{3}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+x^{2}+2 x +1}+x^{2}}{1+x}\right )}{2}\) | \(83\) |
risch | \(-\frac {2 x^{8}+3 x^{7}-x^{6}-11 x^{5}-11 x^{4}-x^{3}+9 x^{2}+8 x +2}{6 \left (1+x \right )^{3} \sqrt {-x^{4}+x^{2}+2 x +1}}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}+\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )+\frac {i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right )}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(1392\) |
default | \(\frac {x \sqrt {-x^{4}+x^{2}+2 x +1}}{3}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}-\frac {\sqrt {-x^{4}+x^{2}+2 x +1}}{2}+\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )+\frac {i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right )}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {5 \sqrt {-x^{4}+x^{2}+2 x +1}}{6 \left (1+x \right )^{2}}+\frac {2 \sqrt {-x^{4}+x^{2}+2 x +1}}{3 \left (1+x \right )}+\frac {\sqrt {-x^{4}+x^{2}+2 x +1}}{3 \left (1+x \right )^{3}}\) | \(1431\) |
elliptic | \(\frac {x \sqrt {-x^{4}+x^{2}+2 x +1}}{3}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )-i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}-\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}-\frac {\sqrt {-x^{4}+x^{2}+2 x +1}}{2}+\frac {i \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (1-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \left (\EllipticF \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )+\frac {i \sqrt {3}\, \EllipticPi \left (\sqrt {\frac {\left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right )}, \sqrt {\frac {\left (-1+\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (-1-\frac {i \sqrt {3}}{2}+\frac {\sqrt {5}}{2}\right ) \left (-1+\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}}\right )}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}\right )}{3 \left (1+\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-\left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {5 \sqrt {-x^{4}+x^{2}+2 x +1}}{6 \left (1+x \right )^{2}}+\frac {2 \sqrt {-x^{4}+x^{2}+2 x +1}}{3 \left (1+x \right )}+\frac {\sqrt {-x^{4}+x^{2}+2 x +1}}{3 \left (1+x \right )^{3}}\) | \(1431\) |
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 {\sqrt {-x^{4} + x^{2} + 2 \, x + 1} {\left (x^{2} + 2 \, x\right )} {\left (x^{2} + x + 1\right )}}{{\left (x + 1\right )}^{4}}\,{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^2+2\,x\right )\,\left (x^2+x+1\right )\,\sqrt {-x^4+x^2+2\,x+1}}{{\left (x+1\right )}^4} \,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 {x \sqrt {- \left (x^{2} - x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right ) \left (x^{2} + x + 1\right )}{\left (x + 1\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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