Optimal. Leaf size=31 \[ \frac {5 \operatorname {EllipticF}\left (\sin ^{-1}(x),-\frac {3}{2}\right )}{3 \sqrt {2}}-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|-\frac {3}{2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {423, 424, 419} \[ \frac {5 F\left (\sin ^{-1}(x)|-\frac {3}{2}\right )}{3 \sqrt {2}}-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|-\frac {3}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 423
Rule 424
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {2+3 x^2}} \, dx &=-\left (\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {1-x^2}} \, dx\right )+\frac {5}{3} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2} E\left (\sin ^{-1}(x)|-\frac {3}{2}\right )+\frac {5 F\left (\sin ^{-1}(x)|-\frac {3}{2}\right )}{3 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.87 \[ -\frac {i E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 0.87 \[ \frac {\left (-2 \EllipticE \left (x , \frac {i \sqrt {6}}{2}\right )+5 \EllipticF \left (x , \frac {i \sqrt {6}}{2}\right )\right ) \sqrt {2}}{6} \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {3\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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