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

Optimal. Leaf size=14 \[ -\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),3\right ) \]

[Out]

-EllipticF[ArcCos[Sqrt[2/3]*x], 3]

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Rubi [A]  time = 0.0116204, antiderivative size = 14, 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} \[ -F\left (\left .\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right )\right |3\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 5*x^2 - 2*x^4],x]

[Out]

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

Mathematica [B]  time = 0.0246362, size = 53, normalized size = 3.79 \[ \frac{\sqrt{3-2 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),\frac{3}{2}\right )}{\sqrt{-4 x^4+10 x^2-6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 - 2*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], 3/2])/Sqrt[-6 + 10*x^2 - 4*x^4]

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Maple [A]  time = 0.051, size = 50, normalized size = 3.6 \begin{align*}{\frac{\sqrt{6}}{6}\sqrt{-6\,{x}^{2}+9}\sqrt{-{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{6}}{3}},{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+5\,{x}^{2}-3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*6^(1/2)*(-6*x^2+9)^(1/2)*(-x^2+1)^(1/2)/(-2*x^4+5*x^2-3)^(1/2)*EllipticF(1/3*x*6^(1/2),1/2*6^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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