Optimal. Leaf size=90 \[ \frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {4 x^4+5 x^2+2}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {4 x^4+5 x^2+2}} \]
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Rubi [A] time = 0.02, antiderivative size = 90, 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 = {1103} \[ \frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {4 x^4+5 x^2+2}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {4 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx &=\frac {\left (1+\sqrt {2} x^2\right ) \sqrt {\frac {2+5 x^2+4 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {2+5 x^2+4 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 147, normalized size = 1.63 \[ -\frac {i \sqrt {1-\frac {8 x^2}{-5-i \sqrt {7}}} \sqrt {1-\frac {8 x^2}{-5+i \sqrt {7}}} F\left (i \sinh ^{-1}\left (2 \sqrt {-\frac {2}{-5-i \sqrt {7}}} x\right )|\frac {-5-i \sqrt {7}}{-5+i \sqrt {7}}\right )}{2 \sqrt {2} \sqrt {-\frac {1}{-5-i \sqrt {7}}} \sqrt {4 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 87, normalized size = 0.97 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {i \sqrt {7}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {i \sqrt {7}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+i \sqrt {7}}\, x}{2}, \frac {\sqrt {9+5 i \sqrt {7}}}{4}\right )}{\sqrt {-5+i \sqrt {7}}\, \sqrt {4 x^{4}+5 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {4\,x^4+5\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {4 x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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