Optimal. Leaf size=35 \[ \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{x^2+3 x+1} \]
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Rubi [F] time = 2.69, 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^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx &=\int \left (\frac {(1-x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{4 \left (1+x+x^2\right )}+\frac {(-3-2 x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{2 \left (1+3 x+x^2\right )^2}+\frac {(1+x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{4 \left (1+3 x+x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {(1-x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {(1+x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \, dx+\frac {1}{2} \int \frac {(-3-2 x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {\left (-1-i \sqrt {3}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x}+\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {3 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2}-\frac {2 x \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx\right )+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx-\int \frac {x \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx\\ &=-\left (\frac {3}{2} \int \left (\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (-3+\sqrt {5}-2 x\right )^2}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x\right )}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (3+\sqrt {5}+2 x\right )^2}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x\right )}\right ) \, dx\right )+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx-\int \left (\frac {2 \left (-3+\sqrt {5}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (-3+\sqrt {5}-2 x\right )^2}-\frac {6 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x\right )}+\frac {2 \left (-3-\sqrt {5}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (3+\sqrt {5}+2 x\right )^2}-\frac {6 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x\right )}\right ) \, dx\\ &=-\left (\frac {6}{5} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (-3+\sqrt {5}-2 x\right )^2} \, dx\right )-\frac {6}{5} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (3+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{5} \left (2 \left (3-\sqrt {5}\right )\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (-3+\sqrt {5}-2 x\right )^2} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{5} \left (2 \left (3+\sqrt {5}\right )\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (3+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.47, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{x^2+3 x+1} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 33, normalized size = 0.94 \begin {gather*} \frac {\sqrt {-x^{4} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1}}{x^{2} + 3 \, x + 1} \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} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 3 \, x + 1\right )}^{2} {\left (x^{2} + x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 31, normalized size = 0.89
method | result | size |
risch | \(-\frac {x^{2}+x +1}{\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}\) | \(31\) |
default | \(\frac {-x^{2}-x -1}{\sqrt {\left (x^{2}+3 x +1\right ) \left (-x^{2}-x -1\right )}}\) | \(33\) |
elliptic | \(\frac {-x^{2}-x -1}{\sqrt {\left (x^{2}+3 x +1\right ) \left (-x^{2}-x -1\right )}}\) | \(33\) |
gosper | \(\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1}\) | \(34\) |
trager | \(\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 31, normalized size = 0.89 \begin {gather*} \frac {\sqrt {x^{2} + x + 1} \sqrt {-x^{2} - 3 \, x - 1}}{x^{2} + 3 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 33, normalized size = 0.94 \begin {gather*} \frac {\sqrt {-x^4-4\,x^3-5\,x^2-4\,x-1}}{x^2+3\,x+1} \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 {- \left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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