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

Optimal. Leaf size=63 \[ \frac{\sqrt{x^2-3} \sqrt{2 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right ),\frac{1}{7}\right )}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}} \]

[Out]

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

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Rubi [A]  time = 0.0064966, antiderivative size = 63, 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} \[ \frac{\sqrt{x^2-3} \sqrt{2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 - 5*x^2 + 2*x^4],x]

[Out]

(Sqrt[-3 + x^2]*Sqrt[1 + 2*x^2]*EllipticF[ArcSin[(Sqrt[7]*x)/Sqrt[-3 + x^2]], 1/7])/(Sqrt[7]*Sqrt[-3 - 5*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{-3-5 x^2+2 x^4}} \, dx &=\frac{\sqrt{-3+x^2} \sqrt{1+2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{-3+x^2}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{-3-5 x^2+2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0249575, size = 65, normalized size = 1.03 \[ -\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{6}\right )}{\sqrt{2} \sqrt{2 x^4-5 x^2-3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.052, size = 53, normalized size = 0.8 \begin{align*}{-{\frac{i}{6}}\sqrt{2}{\it EllipticF} \left ( ix\sqrt{2},{\frac{i}{6}}\sqrt{6} \right ) \sqrt{2\,{x}^{2}+1}\sqrt{-3\,{x}^{2}+9}{\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]

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