3.129 \(\int \frac{1}{\sqrt{2+5 x^2-5 x^4}} \, dx\)

Optimal. Leaf size=48 \[ \sqrt{\frac{2}{\sqrt{65}-5}} F\left (\sin ^{-1}\left (\sqrt{\frac{10}{5+\sqrt{65}}} x\right )|\frac{1}{4} \left (-9-\sqrt{65}\right )\right ) \]

[Out]

Sqrt[2/(-5 + Sqrt[65])]*EllipticF[ArcSin[Sqrt[10/(5 + Sqrt[65])]*x], (-9 - Sqrt[
65])/4]

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Rubi [A]  time = 0.224895, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \sqrt{\frac{2}{\sqrt{65}-5}} F\left (\sin ^{-1}\left (\sqrt{\frac{10}{5+\sqrt{65}}} x\right )|\frac{1}{4} \left (-9-\sqrt{65}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 5*x^2 - 5*x^4],x]

[Out]

Sqrt[2/(-5 + Sqrt[65])]*EllipticF[ArcSin[Sqrt[10/(5 + Sqrt[65])]*x], (-9 - Sqrt[
65])/4]

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Rubi in Sympy [A]  time = 13.3757, size = 51, normalized size = 1.06 \[ - \frac{4 \sqrt{5} F\left (\operatorname{asin}{\left (\frac{x \sqrt{-5 + \sqrt{65}}}{2} \right )}\middle | - \frac{9}{4} - \frac{\sqrt{65}}{4}\right )}{\sqrt{5 + \sqrt{65}} \left (- \sqrt{65} + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-5*x**4+5*x**2+2)**(1/2),x)

[Out]

-4*sqrt(5)*elliptic_f(asin(x*sqrt(-5 + sqrt(65))/2), -9/4 - sqrt(65)/4)/(sqrt(5
+ sqrt(65))*(-sqrt(65) + 5))

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Mathematica [C]  time = 0.100152, size = 52, normalized size = 1.08 \[ -i \sqrt{\frac{2}{5+\sqrt{65}}} F\left (i \sinh ^{-1}\left (\frac{1}{2} \sqrt{5+\sqrt{65}} x\right )|\frac{1}{4} \left (-9+\sqrt{65}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[2 + 5*x^2 - 5*x^4],x]

[Out]

(-I)*Sqrt[2/(5 + Sqrt[65])]*EllipticF[I*ArcSinh[(Sqrt[5 + Sqrt[65]]*x)/2], (-9 +
Sqrt[65])/4]

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Maple [B]  time = 0.117, size = 80, normalized size = 1.7 \[ 2\,{\frac{\sqrt{1- \left ( -5/4+1/4\,\sqrt{65} \right ){x}^{2}}\sqrt{1- \left ( -5/4-1/4\,\sqrt{65} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+\sqrt{65}},i/4\sqrt{10}+i/4\sqrt{26} \right ) }{\sqrt{-5+\sqrt{65}}\sqrt{-5\,{x}^{4}+5\,{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-5*x^4+5*x^2+2)^(1/2),x)

[Out]

2/(-5+65^(1/2))^(1/2)*(1-(-5/4+1/4*65^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*65^(1/2))*x
^2)^(1/2)/(-5*x^4+5*x^2+2)^(1/2)*EllipticF(1/2*x*(-5+65^(1/2))^(1/2),1/4*I*10^(1
/2)+1/4*I*26^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-5 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-5*x^4 + 5*x^2 + 2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-5*x^4 + 5*x^2 + 2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 5 x^{4} + 5 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-5*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-5*x**4 + 5*x**2 + 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-5 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-5*x^4 + 5*x^2 + 2),x, algorithm="giac")

[Out]

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