3.133 \(\int \frac{1}{\sqrt{2+5 x^2-9 x^4}} \, dx\)

Optimal. Leaf size=49 \[ \sqrt{\frac{2}{\sqrt{97}-5}} F\left (\sin ^{-1}\left (3 \sqrt{\frac{2}{5+\sqrt{97}}} x\right )|\frac{1}{36} \left (-61-5 \sqrt{97}\right )\right ) \]

[Out]

Sqrt[2/(-5 + Sqrt[97])]*EllipticF[ArcSin[3*Sqrt[2/(5 + Sqrt[97])]*x], (-61 - 5*S
qrt[97])/36]

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Rubi [A]  time = 0.168765, antiderivative size = 49, 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 \[ \sqrt{\frac{2}{\sqrt{97}-5}} F\left (\sin ^{-1}\left (3 \sqrt{\frac{2}{5+\sqrt{97}}} x\right )|\frac{1}{36} \left (-61-5 \sqrt{97}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 5*x^2 - 9*x^4],x]

[Out]

Sqrt[2/(-5 + Sqrt[97])]*EllipticF[ArcSin[3*Sqrt[2/(5 + Sqrt[97])]*x], (-61 - 5*S
qrt[97])/36]

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Rubi in Sympy [A]  time = 13.8525, size = 48, normalized size = 0.98 \[ - \frac{12 F\left (\operatorname{asin}{\left (\frac{x \sqrt{-5 + \sqrt{97}}}{2} \right )}\middle | - \frac{61}{36} - \frac{5 \sqrt{97}}{36}\right )}{\sqrt{5 + \sqrt{97}} \left (- \sqrt{97} + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-9*x**4+5*x**2+2)**(1/2),x)

[Out]

-12*elliptic_f(asin(x*sqrt(-5 + sqrt(97))/2), -61/36 - 5*sqrt(97)/36)/(sqrt(5 +
sqrt(97))*(-sqrt(97) + 5))

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Mathematica [C]  time = 0.08241, size = 56, normalized size = 1.14 \[ -i \sqrt{\frac{2}{5+\sqrt{97}}} F\left (i \sinh ^{-1}\left (3 \sqrt{\frac{2}{-5+\sqrt{97}}} x\right )|\frac{1}{36} \left (-61+5 \sqrt{97}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[2 + 5*x^2 - 9*x^4],x]

[Out]

(-I)*Sqrt[2/(5 + Sqrt[97])]*EllipticF[I*ArcSinh[3*Sqrt[2/(-5 + Sqrt[97])]*x], (-
61 + 5*Sqrt[97])/36]

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Maple [B]  time = 0.12, size = 80, normalized size = 1.6 \[ 2\,{\frac{\sqrt{1- \left ( -5/4+1/4\,\sqrt{97} \right ){x}^{2}}\sqrt{1- \left ( -5/4-1/4\,\sqrt{97} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+\sqrt{97}},{\frac{5\,i}{12}}\sqrt{2}+i/12\sqrt{194} \right ) }{\sqrt{-5+\sqrt{97}}\sqrt{-9\,{x}^{4}+5\,{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-9*x^4+5*x^2+2)^(1/2),x)

[Out]

2/(-5+97^(1/2))^(1/2)*(1-(-5/4+1/4*97^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*97^(1/2))*x
^2)^(1/2)/(-9*x^4+5*x^2+2)^(1/2)*EllipticF(1/2*x*(-5+97^(1/2))^(1/2),5/12*I*2^(1
/2)+1/12*I*194^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-9 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-9*x^4 + 5*x^2 + 2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-9*x^4 + 5*x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-9 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-9*x^4 + 5*x^2 + 2),x, algorithm="fricas")

[Out]

integral(1/sqrt(-9*x^4 + 5*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 9 x^{4} + 5 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-9*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-9*x**4 + 5*x**2 + 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-9 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-9*x^4 + 5*x^2 + 2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-9*x^4 + 5*x^2 + 2), x)