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

Optimal. Leaf size=40 \[ \frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)
*a)

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Rubi [A]  time = 0.174949, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

Antiderivative was successfully verified.

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

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)
*a)

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Rubi in Sympy [A]  time = 23.7051, size = 42, normalized size = 1.05 \[ \frac{\sqrt{2} \cdot 5^{\frac{3}{4}} \Pi \left (- \frac{2 \sqrt{5} b}{5 a}; \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt [4]{5} x}{2} \right )}\middle | -1\right )}{10 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*5**(3/4)*elliptic_pi(-2*sqrt(5)*b/(5*a), asin(sqrt(2)*5**(1/4)*x/2), -1)
/(10*a)

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Mathematica [A]  time = 0.0628098, size = 43, normalized size = 1.08 \[ -\frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .-\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

Antiderivative was successfully verified.

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

[Out]

-(EllipticPi[(-2*b)/(Sqrt[5]*a), -ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1
/4)*a))

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Maple [B]  time = 0.072, size = 79, normalized size = 2. \[{\frac{\sqrt{2}{5}^{{\frac{3}{4}}}}{5\,a}\sqrt{1-{\frac{\sqrt{5}{x}^{2}}{2}}}\sqrt{1+{\frac{\sqrt{5}{x}^{2}}{2}}}{\it EllipticPi} \left ({\frac{\sqrt [4]{5}x\sqrt{2}}{2}},-{\frac{2\,\sqrt{5}b}{5\,a}},{\frac{\sqrt{-{\frac{\sqrt{5}}{2}}}\sqrt{2}{5}^{{\frac{3}{4}}}}{5}} \right ){\frac{1}{\sqrt{-5\,{x}^{4}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5/a*2^(1/2)*5^(3/4)*(1-1/2*5^(1/2)*x^2)^(1/2)*(1+1/2*5^(1/2)*x^2)^(1/2)/(-5*x^
4+4)^(1/2)*EllipticPi(1/2*5^(1/4)*x*2^(1/2),-2/5*b/a*5^(1/2),1/5*(-1/2*5^(1/2))^
(1/2)*2^(1/2)*5^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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