Optimal. Leaf size=88 \[ -\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}} \]
[Out]
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Rubi [A] time = 0.0922831, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^4]*x^2),x]
[Out]
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Rubi in Sympy [A] time = 5.61417, size = 80, normalized size = 0.91 \[ - \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b/x**4)**(1/2),x)
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Mathematica [C] time = 0.0581153, size = 77, normalized size = 0.88 \[ -\frac{i \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^4]*x^2),x]
[Out]
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Maple [C] time = 0.012, size = 86, normalized size = 1. \[{\frac{1}{{x}^{2}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b/x^4)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.7578, size = 37, normalized size = 0.42 \[ - \frac{\Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} x \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^4)*x^2),x, algorithm="giac")
[Out]