3.340 \(\int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{7} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2]/7

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Rubi [A]  time = 0.109498, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{7} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]

[Out]

EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2]/7

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Rubi in Sympy [A]  time = 17.7466, size = 17, normalized size = 1. \[ \frac{\Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(5*x**2+7)/(-x**4+x**2+2)**(1/2),x)

[Out]

elliptic_pi(-10/7, asin(sqrt(2)*x/2), -2)/7

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Mathematica [C]  time = 0.0443375, size = 24, normalized size = 1.41 \[ -\frac{i \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )}{7 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]

[Out]

((-I/7)*EllipticPi[5/7, I*ArcSinh[x], -1/2])/Sqrt[2]

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Maple [B]  time = 0.018, size = 48, normalized size = 2.8 \[{\frac{\sqrt{2}}{7}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x)

[Out]

1/7*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticPi(1/2*2^
(1/2)*x,-10/7,I*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(5*x**2+7)/(-x**4+x**2+2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x**2 - 2)*(x**2 + 1))*(5*x**2 + 7)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)), x)