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

Optimal. Leaf size=52 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]

[Out]

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

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Rubi [A]  time = 0.007413, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1100} \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 + 3*x^4],x]

[Out]

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

Rule 1100

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

Rubi steps

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

Mathematica [C]  time = 0.0252434, size = 58, normalized size = 1.12 \[ -\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{9 x^4+15 x^2+6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.053, size = 44, normalized size = 0.9 \begin{align*}{-{\frac{i}{2}}\sqrt{{x}^{2}+1}\sqrt{6\,{x}^{2}+4}{\it EllipticF} \left ( ix,{\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*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticF(I*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{1}{\sqrt{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(1/sqrt(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)