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

Optimal. Leaf size=20 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}} \]

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

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Rubi [A]  time = 0.0304281, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

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Rubi in Sympy [A]  time = 4.95618, size = 20, normalized size = 1. \[ \frac{\sqrt{5} F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | - \frac{2}{5}\right )}{5} \]

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

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Mathematica [A]  time = 0.0343294, size = 20, normalized size = 1. \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

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Maple [A]  time = 0.052, size = 25, normalized size = 1.3 \[{\frac{\sqrt{5}}{5}{\it EllipticF} \left ({\frac{\sqrt{5}\sqrt{2}x}{2}},{\frac{i}{5}}\sqrt{5}\sqrt{2} \right ) } \]

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]

1/5*5^(1/2)*EllipticF(1/2*5^(1/2)*2^(1/2)*x,1/5*I*5^(1/2)*2^(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. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}, x\right ) \]

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(1/(sqrt(x^2 + 1)*sqrt(-5*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)