Optimal. Leaf size=63 \[ \frac {\sqrt {x^2-2} \sqrt {3 x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {x^2-2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {3 x^4-5 x^2-2}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, 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 = {1097} \[ \frac {\sqrt {x^2-2} \sqrt {3 x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {x^2-2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-5 x^2+3 x^4}} \, dx &=\frac {\sqrt {-2+x^2} \sqrt {1+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {-2+x^2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {-2-5 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 65, normalized size = 1.03 \[ -\frac {i \sqrt {1-\frac {x^2}{2}} \sqrt {3 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt {3} x\right )|-\frac {1}{6}\right )}{\sqrt {3} \sqrt {3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, 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 [C] time = 0.02, size = 53, normalized size = 0.84 \[ -\frac {i \sqrt {3}\, \sqrt {3 x^{2}+1}\, \sqrt {-2 x^{2}+4}\, \EllipticF \left (i \sqrt {3}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {3 x^{4}-5 x^{2}-2}} \]
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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