Optimal. Leaf size=12 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),-\frac {3}{2}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {419} \[ \frac {F\left (\sin ^{-1}(x)|-\frac {3}{2}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx &=\frac {F\left (\sin ^{-1}(x)|-\frac {3}{2}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),-\frac {3}{2}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}{3 \, x^{4} - x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.17 \[ \frac {\sqrt {2}\, \EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )}{2} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {3\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.31, size = 19, normalized size = 1.58 \[ \begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\relax (x )}\middle | - \frac {3}{2}\right )}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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