3.2085 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

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]

-((Sqrt[b]*Sqrt[a + b/x^4])/(a*(Sqrt[a] + Sqrt[b]/x^2)*x)) + (Sqrt[a + b/x^4]*x)
/a + (b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2
)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b/x^4]) - (b^
(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ellipt
icF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b/x^4])

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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]

-((Sqrt[b]*Sqrt[a + b/x^4])/(a*(Sqrt[a] + Sqrt[b]/x^2)*x)) + (Sqrt[a + b/x^4]*x)
/a + (b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2
)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b/x^4]) - (b^
(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ellipt
icF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[a + b/x^4])

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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]

-sqrt(b)*sqrt(a + b/x**4)/(a*x*(sqrt(a) + sqrt(b)/x**2)) + x*sqrt(a + b/x**4)/a
+ b**(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**
2)*elliptic_e(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(a**(3/4)*sqrt(a + b/x**4)) -
b**(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**2)
*elliptic_f(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(2*a**(3/4)*sqrt(a + b/x**4))

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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]

(I*Sqrt[1 + (a*x^4)/b]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1] -
EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(((I*Sqrt[a])/Sqrt[b])^(
3/2)*Sqrt[a + b/x^4]*x^2)

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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]

I/((a*x^4+b)/x^4)^(1/2)/x^2*b^(1/2)/(I*a^(1/2)/b^(1/2))^(1/2)*(-(I*a^(1/2)*x^2-b
^(1/2))/b^(1/2))^(1/2)*((I*a^(1/2)*x^2+b^(1/2))/b^(1/2))^(1/2)/a^(1/2)*(Elliptic
F(x*(I*a^(1/2)/b^(1/2))^(1/2),I)-EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I))

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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]

integrate(1/sqrt(a + b/x^4), x)

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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]

integral(1/sqrt((a*x^4 + b)/x^4), x)

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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]

-x*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*exp_polar(I*pi)/(a*x**4))/(4*sqrt(a)
*gamma(3/4))

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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]

integrate(1/sqrt(a + b/x^4), x)