Optimal. Leaf size=54 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]
[Out]
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Rubi [A] time = 0.145094, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{3}}\right )\right |-1\right )}{b^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[3 - b*x^4],x]
[Out]
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Rubi in Sympy [A] time = 23.8078, size = 56, normalized size = 1.04 \[ \frac{\sqrt [4]{3} E\left (\operatorname{asin}{\left (\frac{3^{\frac{3}{4}} \sqrt [4]{b} x}{3} \right )}\middle | -1\right )}{b^{\frac{3}{4}}} - \frac{\sqrt [4]{3} F\left (\operatorname{asin}{\left (\frac{3^{\frac{3}{4}} \sqrt [4]{b} x}{3} \right )}\middle | -1\right )}{b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(-b*x**4+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0705489, size = 76, normalized size = 1.41 \[ \frac{i \sqrt [4]{3} \sqrt{-\sqrt{b}} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b}} x}{\sqrt [4]{3}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b}} x}{\sqrt [4]{3}}\right )\right |-1\right )\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[3 - b*x^4],x]
[Out]
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Maple [B] time = 0.02, size = 94, normalized size = 1.7 \[ -{\frac{1}{3}\sqrt{9-3\,\sqrt{3}\sqrt{b}{x}^{2}}\sqrt{9+3\,\sqrt{3}\sqrt{b}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{\sqrt{3}\sqrt{b}}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{\sqrt{3}\sqrt{b}}},i \right ) \right ){\frac{1}{\sqrt{\sqrt{3}\sqrt{b}}}}{\frac{1}{\sqrt{-b{x}^{4}+3}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(-b*x^4+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-b x^{4} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.1119, size = 39, normalized size = 0.72 \[ \frac{\sqrt{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(-b*x**4+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-b x^{4} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-b*x^4 + 3),x, algorithm="giac")
[Out]