3.80 \(\int \frac{1}{\sqrt{3+7 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\sqrt{\frac{x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{5}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]

[Out]

(Sqrt[(3 + x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*EllipticF[ArcTan[Sqrt[2]*x], 5/6])/(Sqr
t[6]*Sqrt[3 + 7*x^2 + 2*x^4])

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Rubi [A]  time = 0.0275477, antiderivative size = 60, 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 \[ \frac{\sqrt{\frac{x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{5}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 + 7*x^2 + 2*x^4],x]

[Out]

(Sqrt[(3 + x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*EllipticF[ArcTan[Sqrt[2]*x], 5/6])/(Sqr
t[6]*Sqrt[3 + 7*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 3.69853, size = 56, normalized size = 0.93 \[ \frac{\sqrt{2} \sqrt{\frac{2 x^{2} + 6}{12 x^{2} + 6}} \left (12 x^{2} + 6\right ) F\left (\operatorname{atan}{\left (\sqrt{2} x \right )}\middle | \frac{5}{6}\right )}{12 \sqrt{2 x^{4} + 7 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4+7*x**2+3)**(1/2),x)

[Out]

sqrt(2)*sqrt((2*x**2 + 6)/(12*x**2 + 6))*(12*x**2 + 6)*elliptic_f(atan(sqrt(2)*x
), 5/6)/(12*sqrt(2*x**4 + 7*x**2 + 3))

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Mathematica [C]  time = 0.0422, size = 61, normalized size = 1.02 \[ -\frac{i \sqrt{x^2+3} \sqrt{2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{2} x\right )|\frac{1}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3 + 7*x^2 + 2*x^4],x]

[Out]

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

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Maple [C]  time = 0.105, size = 50, normalized size = 0.8 \[{-{\frac{i}{3}}\sqrt{3}{\it EllipticF} \left ({\frac{i}{3}}\sqrt{3}x,\sqrt{6} \right ) \sqrt{3\,{x}^{2}+9}\sqrt{2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+7\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4+7*x^2+3)^(1/2),x)

[Out]

-1/3*I*3^(1/2)*(3*x^2+9)^(1/2)*(2*x^2+1)^(1/2)/(2*x^4+7*x^2+3)^(1/2)*EllipticF(1
/3*I*3^(1/2)*x,6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 7*x^2 + 3),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 7*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 7*x^2 + 3),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 + 7*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} + 7 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*x**4+7*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 7*x**2 + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 + 7*x^2 + 3),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 7*x^2 + 3), x)