Optimal. Leaf size=60 \[ \frac {\sqrt {\frac {x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt {2} x\right )|\frac {5}{6}\right )}{\sqrt {6} \sqrt {2 x^4+7 x^2+3}} \]
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Rubi [A] time = 0.01, antiderivative size = 60, 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 = {1099} \[ \frac {\sqrt {\frac {x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt {2} x\right )|\frac {5}{6}\right )}{\sqrt {6} \sqrt {2 x^4+7 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+7 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {3+x^2}{1+2 x^2}} \left (1+2 x^2\right ) F\left (\tan ^{-1}\left (\sqrt {2} x\right )|\frac {5}{6}\right )}{\sqrt {6} \sqrt {3+7 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 61, normalized size = 1.02 \[ -\frac {i \sqrt {x^2+3} \sqrt {2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{6}\right )}{\sqrt {6} \sqrt {2 x^4+7 x^2+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} + 3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 50, normalized size = 0.83 \[ -\frac {i \sqrt {3}\, \sqrt {3 x^{2}+9}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (\frac {i \sqrt {3}\, x}{3}, \sqrt {6}\right )}{3 \sqrt {2 x^{4}+7 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {2\,x^4+7\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 x^{4} + 7 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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