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

Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-5 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right ),\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4-5 x^2+2}} \]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - 5*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(3/2)^(1/4)*x], (12 + 5*Sq
rt[6])/24])/(2*6^(1/4)*Sqrt[2 - 5*x^2 + 3*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0151634, antiderivative size = 92, 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 = {1096} \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-5 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4-5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 - 5*x^2 + 3*x^4],x]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - 5*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[(3/2)^(1/4)*x], (12 + 5*Sq
rt[6])/24])/(2*6^(1/4)*Sqrt[2 - 5*x^2 + 3*x^4])

Rule 1096

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2-5 x^2+3 x^4}} \, dx &=\frac{\left (2+\sqrt{6} x^2\right ) \sqrt{\frac{2-5 x^2+3 x^4}{\left (2+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2-5 x^2+3 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.022652, size = 53, normalized size = 0.58 \[ \frac{\sqrt{2-3 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{9 x^4-15 x^2+6}} \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 42, normalized size = 0.5 \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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)