Optimal. Leaf size=49 \[ \sqrt {\frac {2}{\sqrt {73}-5}} F\left (\sin ^{-1}\left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt {\frac {2}{\sqrt {73}-5}} F\left (\sin ^{-1}\left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-6 x^4}} \, dx &=\left (2 \sqrt {6}\right ) \int \frac {1}{\sqrt {5+\sqrt {73}-12 x^2} \sqrt {-5+\sqrt {73}+12 x^2}} \, dx\\ &=\sqrt {\frac {2}{-5+\sqrt {73}}} F\left (\sin ^{-1}\left (2 \sqrt {\frac {3}{5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49-5 \sqrt {73}\right )\right )\\ \end {align*}
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Mathematica [C] time = 0.05, size = 56, normalized size = 1.14 \[ -i \sqrt {\frac {2}{5+\sqrt {73}}} F\left (i \sinh ^{-1}\left (2 \sqrt {\frac {3}{-5+\sqrt {73}}} x\right )|\frac {1}{24} \left (-49+5 \sqrt {73}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-6 \, x^{4} + 5 \, x^{2} + 2}}{6 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-6 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 80, normalized size = 1.63 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {\sqrt {73}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {\sqrt {73}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+\sqrt {73}}\, x}{2}, \frac {5 i \sqrt {3}}{12}+\frac {i \sqrt {219}}{12}\right )}{\sqrt {-5+\sqrt {73}}\, \sqrt {-6 x^{4}+5 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {-6 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-6\,x^4+5\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 6 x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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