Optimal. Leaf size=35 \[ \frac{11 F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{6 \sqrt{2}}-\frac{2}{3} \sqrt{2} E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right ) \]
[Out]
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Rubi [A] time = 0.0671248, antiderivative size = 35, 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 \[ \frac{11 F\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right )}{6 \sqrt{2}}-\frac{2}{3} \sqrt{2} E\left (\sin ^{-1}(2 x)|-\frac{3}{8}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 12.4344, size = 36, normalized size = 1.03 \[ - \frac{2 \sqrt{2} E\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{3} + \frac{11 \sqrt{2} F\left (\operatorname{asin}{\left (2 x \right )}\middle | - \frac{3}{8}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0263359, size = 27, normalized size = 0.77 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{8}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 4*x^2]/Sqrt[2 + 3*x^2],x]
[Out]
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Maple [A] time = 0.029, size = 37, normalized size = 1.1 \[{\frac{ \left ( 11\,{\it EllipticF} \left ( 2\,x,i/4\sqrt{3}\sqrt{2} \right ) -8\,{\it EllipticE} \left ( 2\,x,i/4\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{2}}{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (2 x - 1\right ) \left (2 x + 1\right )}}{\sqrt{3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="giac")
[Out]