Optimal. Leaf size=128 \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]
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Rubi [A] time = 0.233451, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{\sqrt{a-b x^4}}{a x}+\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}-\frac{\sqrt [4]{b} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a - b*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 39.9791, size = 110, normalized size = 0.86 \[ - \frac{\sqrt{a - b x^{4}}}{a x} - \frac{\sqrt [4]{b} \sqrt{1 - \frac{b x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{b} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(-b*x**4+a)**(1/2),x)
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Mathematica [C] time = 0.490198, size = 115, normalized size = 0.9 \[ \frac{-i \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{1-\frac{b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )+\frac{b x^3}{a}-\frac{1}{x}}{\sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a - b*x^4]),x]
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Maple [A] time = 0.015, size = 106, normalized size = 0.8 \[ -{\frac{1}{ax}\sqrt{-b{x}^{4}+a}}+{1\sqrt{b}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(-b*x^4+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x^4 + a)*x^2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-b x^{4} + a} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x^4 + a)*x^2),x, algorithm="fricas")
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Sympy [A] time = 2.46498, size = 41, normalized size = 0.32 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(-b*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x^4 + a)*x^2),x, algorithm="giac")
[Out]