3.109 \(\int \frac{1}{\sqrt{-2+5 x^2-3 x^4}} \, dx\)

Optimal. Leaf size=6 \[ -\text{EllipticF}\left (\cos ^{-1}(x),3\right ) \]

[Out]

-EllipticF[ArcCos[x], 3]

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Rubi [A]  time = 0.0104999, antiderivative size = 6, 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, Rules used = {1095, 420} \[ -F\left (\left .\cos ^{-1}(x)\right |3\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 5*x^2 - 3*x^4],x]

[Out]

-EllipticF[ArcCos[x], 3]

Rule 1095

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rule 420

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> -Simp[EllipticF[ArcCos[Rt[-(d/c), 2]
*x], (b*c)/(b*c - a*d)]/(Sqrt[c]*Rt[-(d/c), 2]*Sqrt[a - (b*c)/d]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &
& GtQ[c, 0] && GtQ[a - (b*c)/d, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2+5 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{6-6 x^2} \sqrt{-4+6 x^2}} \, dx\\ &=-F\left (\left .\cos ^{-1}(x)\right |3\right )\\ \end{align*}

Mathematica [B]  time = 0.0264863, size = 53, normalized size = 8.83 \[ \frac{\sqrt{2-3 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{-9 x^4+15 x^2-6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 + 5*x^2 - 3*x^4],x]

[Out]

(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqrt[-6 + 15*x^2 - 9*x^4]

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Maple [A]  time = 0.049, size = 42, normalized size = 7. \begin{align*}{\frac{1}{2}\sqrt{-{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\it EllipticF} \left ( x,{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{-3\,{x}^{4}+5\,{x}^{2}-2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-x^2+1)^(1/2)*(-6*x^2+4)^(1/2)/(-3*x^4+5*x^2-2)^(1/2)*EllipticF(x,1/2*6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+5*x^2-2)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+5*x^2-2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{4} + 5 x^{2} - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**4+5*x**2-2)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^4+5*x^2-2)^(1/2),x, algorithm="giac")

[Out]

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