Optimal. Leaf size=52 \[ -\frac {\sqrt {-3 x^2-2} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {3 x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.02, antiderivative size = 52, 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, 418} \[ -\frac {\sqrt {-3 x^2-2} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {3 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-5 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-4-6 x^2} \sqrt {6+6 x^2}} \, dx\\ &=-\frac {\sqrt {-2-3 x^2} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {2+3 x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 63, normalized size = 1.21 \[ -\frac {i \sqrt {x^2+1} \sqrt {3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {3} \sqrt {-3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} - 5 \, x^{2} - 2}}{3 \, x^{4} + 5 \, x^{2} + 2}, x\right ) \]
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 = 0.01, size = 50, normalized size = 0.96 \[ -\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {6}\, x}{2}, \frac {\sqrt {6}}{3}\right )}{6 \sqrt {-3 x^{4}-5 x^{2}-2}} \]
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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