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

Optimal. Leaf size=53 \[ \frac{\sqrt{2 x^2+3} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{3}\right )}{\sqrt{3} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+3}{x^2+1}}} \]

[Out]

(Sqrt[3 + 2*x^2]*EllipticF[ArcTan[x], 1/3])/(Sqrt[3]*Sqrt[-1 - x^2]*Sqrt[(3 + 2*x^2)/(1 + x^2)])

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Rubi [A]  time = 0.0161595, antiderivative size = 53, 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, 418} \[ \frac{\sqrt{2 x^2+3} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+3}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 - 5*x^2 - 2*x^4],x]

[Out]

(Sqrt[3 + 2*x^2]*EllipticF[ArcTan[x], 1/3])/(Sqrt[3]*Sqrt[-1 - x^2]*Sqrt[(3 + 2*x^2)/(1 + x^2)])

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 418

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

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

Mathematica [C]  time = 0.0243654, size = 63, normalized size = 1.19 \[ -\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),\frac{3}{2}\right )}{\sqrt{2} \sqrt{-2 x^4-5 x^2-3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 + x^2]*Sqrt[3 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/3]*x], 3/2])/(Sqrt[2]*Sqrt[-3 - 5*x^2 - 2*x^4])

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Maple [C]  time = 0.053, size = 44, normalized size = 0.8 \begin{align*}{-{\frac{i}{3}}\sqrt{{x}^{2}+1}\sqrt{6\,{x}^{2}+9}{\it EllipticF} \left ( ix,{\frac{\sqrt{6}}{3}} \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]

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