Optimal. Leaf size=92 \[ \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-5 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-5 x^2+2 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 92, 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 = {1110}
\begin {gather*} \frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4-5 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2 x^4-5 x^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1110
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-5 x^2+2 x^4}} \, dx &=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3-5 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{24} \left (12+5 \sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3-5 x^2+2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.02, size = 53, normalized size = 0.58 \begin {gather*} \frac {\sqrt {3-2 x^2} \sqrt {1-x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|\frac {3}{2}\right )}{\sqrt {6-10 x^2+4 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 42, normalized size = 0.46
method | result | size |
default | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \EllipticF \left (x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {2 x^{4}-5 x^{2}+3}}\) | \(42\) |
elliptic | \(\frac {\sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+9}\, \EllipticF \left (x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {2 x^{4}-5 x^{2}+3}}\) | \(42\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 8, normalized size = 0.09 \begin {gather*} \frac {1}{3} \, \sqrt {3} {\rm ellipticF}\left (x, \frac {2}{3}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {2 x^{4} - 5 x^{2} + 3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {2\,x^4-5\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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