Optimal. Leaf size=67 \[ \frac {\sqrt {x^2+3} \sqrt {2 x^2-1} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{3}} x}{\sqrt {2 x^2-1}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {2 x^4+5 x^2-3}} \]
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Rubi [A] time = 0.01, antiderivative size = 67, 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} \[ \frac {\sqrt {x^2+3} \sqrt {2 x^2-1} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{3}} x}{\sqrt {2 x^2-1}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {2 x^4+5 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+5 x^2+2 x^4}} \, dx &=\frac {\sqrt {3+x^2} \sqrt {-1+2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{3}} x}{\sqrt {-1+2 x^2}}\right )|\frac {6}{7}\right )}{\sqrt {7} \sqrt {-3+5 x^2+2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.81 \[ \frac {\sqrt {1-2 x^2} \sqrt {x^2+3} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {6} \sqrt {2 x^4+5 x^2-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 5 \, x^{2} - 3}}, x\right ) \]
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 [C] time = 0.01, size = 53, normalized size = 0.79 \[ -\frac {i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {3}\, x}{3}, i \sqrt {6}\right )}{3 \sqrt {2 x^{4}+5 x^{2}-3}} \]
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 = -0.01 \[ \int \frac {1}{\sqrt {2\,x^4+5\,x^2-3}} \,d x \]
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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