Optimal. Leaf size=10 \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-6\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-6\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {12-4 x^2} \sqrt {2+4 x^2}} \, dx\\ &=F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-6\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 65, normalized size = 6.50 \[ -\frac {i \sqrt {1-\frac {x^2}{3}} \sqrt {2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {2} \sqrt {-2 x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 5 \, x^{2} + 3}}{2 \, x^{4} - 5 \, x^{2} - 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 51, normalized size = 5.10 \[ \frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {3}\, x}{3}, i \sqrt {6}\right )}{3 \sqrt {-2 x^{4}+5 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {-2\,x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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