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

Optimal. Leaf size=67 \[ \frac{\sqrt{x^2+2} \sqrt{3 x^2-1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}} x}{\sqrt{3 x^2-1}}\right ),\frac{6}{7}\right )}{\sqrt{7} \sqrt{3 x^4+5 x^2-2}} \]

[Out]

(Sqrt[2 + x^2]*Sqrt[-1 + 3*x^2]*EllipticF[ArcSin[(Sqrt[7/2]*x)/Sqrt[-1 + 3*x^2]], 6/7])/(Sqrt[7]*Sqrt[-2 + 5*x
^2 + 3*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0082686, antiderivative size = 67, 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 = {1097} \[ \frac{\sqrt{x^2+2} \sqrt{3 x^2-1} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}} x}{\sqrt{3 x^2-1}}\right )|\frac{6}{7}\right )}{\sqrt{7} \sqrt{3 x^4+5 x^2-2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 5*x^2 + 3*x^4],x]

[Out]

(Sqrt[2 + x^2]*Sqrt[-1 + 3*x^2]*EllipticF[ArcSin[(Sqrt[7/2]*x)/Sqrt[-1 + 3*x^2]], 6/7])/(Sqrt[7]*Sqrt[-2 + 5*x
^2 + 3*x^4])

Rule 1097

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2+5 x^2+3 x^4}} \, dx &=\frac{\sqrt{2+x^2} \sqrt{-1+3 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}} x}{\sqrt{-1+3 x^2}}\right )|\frac{6}{7}\right )}{\sqrt{7} \sqrt{-2+5 x^2+3 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0219737, size = 54, normalized size = 0.81 \[ \frac{\sqrt{1-3 x^2} \sqrt{x^2+2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} x\right ),-\frac{1}{6}\right )}{\sqrt{6} \sqrt{3 x^4+5 x^2-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.052, size = 53, normalized size = 0.8 \begin{align*}{-{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},i\sqrt{6} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\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*2^(1/2)*(2*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)/(3*x^4+5*x^2-2)^(1/2)*EllipticF(1/2*I*x*2^(1/2),I*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)