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

Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{73}-7}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]

[Out]

Sqrt[2/(-7 + Sqrt[73])]*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqr
t[73])/12]

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Rubi [A]  time = 0.142173, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \sqrt{\frac{2}{\sqrt{73}-7}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[2/(-7 + Sqrt[73])]*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqr
t[73])/12]

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Rubi in Sympy [A]  time = 12.8836, size = 58, normalized size = 1.29 \[ - \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (\frac{\sqrt{6} x \sqrt{-7 + \sqrt{73}}}{6} \right )}\middle | - \frac{61}{12} - \frac{7 \sqrt{73}}{12}\right )}{\sqrt{7 + \sqrt{73}} \left (- \sqrt{73} + 7\right )} \]

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]

-4*sqrt(3)*elliptic_f(asin(sqrt(6)*x*sqrt(-7 + sqrt(73))/6), -61/12 - 7*sqrt(73)
/12)/(sqrt(7 + sqrt(73))*(-sqrt(73) + 7))

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Mathematica [C]  time = 0.0760104, size = 52, normalized size = 1.16 \[ -i \sqrt{\frac{2}{7+\sqrt{73}}} F\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{-7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61+7 \sqrt{73}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-I)*Sqrt[2/(7 + Sqrt[73])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-7 + Sqrt[73]]], (-61
+ 7*Sqrt[73])/12]

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Maple [B]  time = 0.117, size = 84, normalized size = 1.9 \[ 6\,{\frac{\sqrt{1- \left ( -7/6+1/6\,\sqrt{73} \right ){x}^{2}}\sqrt{1- \left ( -1/6\,\sqrt{73}-7/6 \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-42+6\,\sqrt{73}},{\frac{7\,i}{12}}\sqrt{6}+i/12\sqrt{438} \right ) }{\sqrt{-42+6\,\sqrt{73}}\sqrt{-2\,{x}^{4}+7\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4+7*x^2+3)^(1/2),x)

[Out]

6/(-42+6*73^(1/2))^(1/2)*(1-(-7/6+1/6*73^(1/2))*x^2)^(1/2)*(1-(-1/6*73^(1/2)-7/6
)*x^2)^(1/2)/(-2*x^4+7*x^2+3)^(1/2)*EllipticF(1/6*x*(-42+6*73^(1/2))^(1/2),7/12*
I*6^(1/2)+1/12*I*438^(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)