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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]