Optimal. Leaf size=52 \[ \frac {\left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}} \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1100} \[ \frac {\left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1100
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx &=\frac {\left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 58, normalized size = 1.12 \[ -\frac {i \sqrt {x^2+1} \sqrt {3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {9 x^4+15 x^2+6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 44, normalized size = 0.85 \[ -\frac {i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \EllipticF \left (i x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {3\,x^4+5\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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