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

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

[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 = 0.0285873, 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 \[ \frac{\sqrt{x^2-3} \sqrt{2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-3}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{2 x^4-5 x^2-3}} \]

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])

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Rubi in Sympy [A]  time = 3.70724, size = 70, normalized size = 1.11 \[ \frac{\sqrt{3} \sqrt{\frac{2 x^{2}}{7} - \frac{6}{7}} \sqrt{12 x^{2} + 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{2 x^{2}}{7} - \frac{6}{7}}} \right )}\middle | \frac{1}{7}\right )}{6 \sqrt{2 x^{4} - 5 x^{2} - 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4-5*x**2-3)**(1/2),x)

[Out]

sqrt(3)*sqrt(2*x**2/7 - 6/7)*sqrt(12*x**2 + 6)*elliptic_f(asin(sqrt(2)*x/sqrt(2*
x**2/7 - 6/7)), 1/7)/(6*sqrt(2*x**4 - 5*x**2 - 3))

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Mathematica [C]  time = 0.0416615, size = 65, normalized size = 1.03 \[ -\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{6}\right )}{\sqrt{2} \sqrt{2 x^4-5 x^2-3}} \]

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])/(Sq
rt[2]*Sqrt[-3 - 5*x^2 + 2*x^4])

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Maple [C]  time = 0.022, size = 53, normalized size = 0.8 \[{-{\frac{i}{6}}\sqrt{2}{\it EllipticF} \left ( i\sqrt{2}x,{\frac{i}{6}}\sqrt{6} \right ) \sqrt{2\,{x}^{2}+1}\sqrt{-3\,{x}^{2}+9}{\frac{1}{\sqrt{2\,{x}^{4}-5\,{x}^{2}-3}}}} \]

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*2^(1/2)*x,1/6*I*6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 5*x^2 - 3),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. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 5*x^2 - 3),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. \[ \int \frac{1}{\sqrt{2 x^{4} - 5 x^{2} - 3}}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 5 \, x^{2} - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 5*x^2 - 3),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 - 5*x^2 - 3), x)