3.131 \(\int \frac{1}{\sqrt{2+5 x^2-7 x^4}} \, dx\)

Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{7}{2}\right )}{\sqrt{2}} \]

[Out]

EllipticF[ArcSin[x], -7/2]/Sqrt[2]

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Rubi [A]  time = 0.042135, antiderivative size = 12, 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 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{7}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 5*x^2 - 7*x^4],x]

[Out]

EllipticF[ArcSin[x], -7/2]/Sqrt[2]

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Rubi in Sympy [A]  time = 10.0808, size = 14, normalized size = 1.17 \[ \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{7}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-7*x**4+5*x**2+2)**(1/2),x)

[Out]

sqrt(2)*elliptic_f(asin(x), -7/2)/2

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Mathematica [C]  time = 0.0444309, size = 65, normalized size = 5.42 \[ -\frac{i \sqrt{1-x^2} \sqrt{7 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{\frac{7}{2}} x\right )|-\frac{2}{7}\right )}{\sqrt{7} \sqrt{-7 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + 5*x^2 - 7*x^4],x]

[Out]

((-I)*Sqrt[1 - x^2]*Sqrt[2 + 7*x^2]*EllipticF[I*ArcSinh[Sqrt[7/2]*x], -2/7])/(Sq
rt[7]*Sqrt[2 + 5*x^2 - 7*x^4])

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Maple [B]  time = 0.018, size = 43, normalized size = 3.6 \[{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{14} \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{14\,{x}^{2}+4}{\frac{1}{\sqrt{-7\,{x}^{4}+5\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-7*x^4+5*x^2+2)^(1/2),x)

[Out]

1/2*(-x^2+1)^(1/2)*(14*x^2+4)^(1/2)/(-7*x^4+5*x^2+2)^(1/2)*EllipticF(x,1/2*I*14^
(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-7 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-7*x^4 + 5*x^2 + 2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-7*x^4 + 5*x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-7 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-7*x^4 + 5*x^2 + 2),x, algorithm="fricas")

[Out]

integral(1/sqrt(-7*x^4 + 5*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 7 x^{4} + 5 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-7*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-7*x**4 + 5*x**2 + 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-7 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-7*x^4 + 5*x^2 + 2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-7*x^4 + 5*x^2 + 2), x)