3.217 \(\int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}} \]

[Out]

EllipticF[ArcSin[x], -3/2]/Sqrt[2]

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Rubi [A]  time = 0.0248665, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

EllipticF[ArcSin[x], -3/2]/Sqrt[2]

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Rubi in Sympy [A]  time = 5.2395, size = 14, normalized size = 1.17 \[ \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{3}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

sqrt(2)*elliptic_f(asin(x), -3/2)/2

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Mathematica [A]  time = 0.0311283, size = 12, normalized size = 1. \[ \frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

EllipticF[ArcSin[x], -3/2]/Sqrt[2]

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Maple [A]  time = 0.025, size = 17, normalized size = 1.4 \[{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{3}\sqrt{2} \right ) \sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+1)^(1/2)/(3*x^2+2)^(1/2),x)

[Out]

1/2*EllipticF(x,1/2*I*3^(1/2)*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 1)*(x + 1))*sqrt(3*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(3*x^2 + 2)*sqrt(-x^2 + 1)), x)