3.254 \(\int \frac{1}{\sqrt{2-5 x^2} \sqrt{-1-x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], -2/5])/(Sqrt[5]*Sqrt[-1 - x^2])

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Rubi [A]  time = 0.0522014, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], -2/5])/(Sqrt[5]*Sqrt[-1 - x^2])

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Rubi in Sympy [A]  time = 8.90966, size = 39, normalized size = 0.98 \[ \frac{\sqrt{5} \sqrt{x^{2} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | - \frac{2}{5}\right )}{5 \sqrt{- x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-5*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)

[Out]

sqrt(5)*sqrt(x**2 + 1)*elliptic_f(asin(sqrt(10)*x/2), -2/5)/(5*sqrt(-x**2 - 1))

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Mathematica [A]  time = 0.0377513, size = 40, normalized size = 1. \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[5/2]*x], -2/5])/(Sqrt[5]*Sqrt[-1 - x^2])

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Maple [A]  time = 0.035, size = 37, normalized size = 0.9 \[{{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{2}\sqrt{5} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-5*x^2+2)^(1/2)/(-x^2-1)^(1/2),x)

[Out]

1/2*I*EllipticF(I*x,1/2*I*2^(1/2)*5^(1/2))*2^(1/2)/(x^2+1)^(1/2)*(-x^2-1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}{5 \, x^{4} + 3 \, x^{2} - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="fricas")

[Out]

integral(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)/(5*x^4 + 3*x^2 - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 5 x^{2} + 2} \sqrt{- x^{2} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-5*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt(-5*x**2 + 2)*sqrt(-x**2 - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)), x)