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

Optimal. Leaf size=63 \[ \frac{\sqrt{x^2-1} \sqrt{2 x^2+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{x^2-1}}\right ),\frac{3}{5}\right )}{\sqrt{5} \sqrt{2 x^4+x^2-3}} \]

[Out]

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

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Rubi [A]  time = 0.0074047, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1097} \[ \frac{\sqrt{x^2-1} \sqrt{2 x^2+3} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{x^2-1}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{2 x^4+x^2-3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + x^2 + 2*x^4],x]

[Out]

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

Mathematica [C]  time = 0.0223485, size = 63, normalized size = 1. \[ -\frac{i \sqrt{1-x^2} \sqrt{2 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),-\frac{3}{2}\right )}{\sqrt{2} \sqrt{2 x^4+x^2-3}} \]

Antiderivative was successfully verified.

[In]

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

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Maple [C]  time = 0.067, size = 51, normalized size = 0.8 \begin{align*}{-{\frac{i}{6}}\sqrt{6}{\it EllipticF} \left ({\frac{i}{3}}x\sqrt{6},{\frac{i}{2}}\sqrt{6} \right ) \sqrt{6\,{x}^{2}+9}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+{x}^{2}-3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*I*6^(1/2)*(6*x^2+9)^(1/2)*(-x^2+1)^(1/2)/(2*x^4+x^2-3)^(1/2)*EllipticF(1/3*I*x*6^(1/2),1/2*I*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} + x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 + x^2 - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + x^{2} - 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(2*x^4 + 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} + x^{2} - 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(2*x**4 + 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} + x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(2*x^4 + x^2 - 3), x)