Optimal. Leaf size=31 \[ \frac{5 \text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right )}{3 \sqrt{2}}-\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]
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Rubi [A] time = 0.0202918, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 423
Rule 424
Rule 419
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*}
Mathematica [C] time = 0.0050336, 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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Maple [A] time = 0.014, size = 27, normalized size = 0.9 \begin{align*}{\frac{ \left ( 5\,{\it EllipticF} \left ( x,i/2\sqrt{6} \right ) -2\,{\it EllipticE} \left ( x,i/2\sqrt{6} \right ) \right ) \sqrt{2}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{3 x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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