Optimal. Leaf size=231 \[ -\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x \sqrt{a+\frac{b}{x^4}}}{a}-\frac{\sqrt{b} \sqrt{a+\frac{b}{x^4}}}{a x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]
[Out]
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Rubi [A] time = 0.296741, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{a^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x \sqrt{a+\frac{b}{x^4}}}{a}-\frac{\sqrt{b} \sqrt{a+\frac{b}{x^4}}}{a x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a + b/x^4],x]
[Out]
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Rubi in Sympy [A] time = 23.5094, size = 202, normalized size = 0.87 \[ - \frac{\sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{a x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} + \frac{x \sqrt{a + \frac{b}{x^{4}}}}{a} + \frac{\sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{\sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0945284, size = 107, normalized size = 0.46 \[ \frac{i \sqrt{\frac{a x^4}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{x^2 \left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a + b/x^4],x]
[Out]
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Maple [C] time = 0.014, size = 113, normalized size = 0.5 \[{\frac{i}{{x}^{2}}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^4)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a + b/x^4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a + b/x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.05644, size = 41, normalized size = 0.18 \[ - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a + b/x^4),x, algorithm="giac")
[Out]