Optimal. Leaf size=108 \[ \frac {\sqrt {\frac {4+\left (5-\sqrt {17}\right ) x^2}{4+\left (5+\sqrt {17}\right ) x^2}} \left (4+\left (5+\sqrt {17}\right ) x^2\right ) F\left (\tan ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {2+5 x^2+x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1113}
\begin {gather*} \frac {\sqrt {\frac {\left (5-\sqrt {17}\right ) x^2+4}{\left (5+\sqrt {17}\right ) x^2+4}} \left (\left (5+\sqrt {17}\right ) x^2+4\right ) F\left (\text {ArcTan}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {x^4+5 x^2+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1113
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2+x^4}} \, dx &=\frac {\sqrt {\frac {4+\left (5-\sqrt {17}\right ) x^2}{4+\left (5+\sqrt {17}\right ) x^2}} \left (4+\left (5+\sqrt {17}\right ) x^2\right ) F\left (\tan ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {2+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.06, size = 103, normalized size = 0.95 \begin {gather*} -\frac {i \sqrt {5-\sqrt {17}+2 x^2} \sqrt {5+\sqrt {17}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {17}}} x\right )|\frac {21}{4}+\frac {5 \sqrt {17}}{4}\right )}{\sqrt {2 \left (5-\sqrt {17}\right )} \sqrt {2+5 x^2+x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.70
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {17}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {17}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-5+\sqrt {17}}}{2}, \frac {5 \sqrt {2}}{4}+\frac {\sqrt {34}}{4}\right )}{\sqrt {-5+\sqrt {17}}\, \sqrt {x^{4}+5 x^{2}+2}}\) | \(76\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {17}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {17}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-5+\sqrt {17}}}{2}, \frac {5 \sqrt {2}}{4}+\frac {\sqrt {34}}{4}\right )}{\sqrt {-5+\sqrt {17}}\, \sqrt {x^{4}+5 x^{2}+2}}\) | \(76\) |
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 [A]
time = 0.10, size = 40, normalized size = 0.37 \begin {gather*} -\frac {1}{8} \, {\left (\sqrt {17} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {17} - 5} {\rm ellipticF}\left (\frac {1}{2} \, x \sqrt {\sqrt {17} - 5}, \frac {5}{4} \, \sqrt {17} + \frac {21}{4}\right ) \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 {1}{\sqrt {x^{4} + 5 x^{2} + 2}}\, 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 {1}{\sqrt {x^4+5\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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