3.2086 \(\int \frac{1}{\sqrt{a+\frac{b}{x^4}} x^2} \, dx\)

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]

-(Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[
2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

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]

-(Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[
2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

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)

[Out]

-sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**2)*ellipti
c_f(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(2*a**(1/4)*b**(1/4)*sqrt(a + b/x**4))

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

1/((a*x^4+b)/x^4)^(1/2)/x^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)*EllipticF(x*(I*a^(1/2)/b^
(1/2))^(1/2),I)

_______________________________________________________________________________________

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]

integrate(1/(sqrt(a + b/x^4)*x^2), x)

_______________________________________________________________________________________

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]

integral(1/(x^2*sqrt((a*x^4 + b)/x^4)), x)

_______________________________________________________________________________________

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]

-gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*sqrt(a)*x*g
amma(5/4))

_______________________________________________________________________________________

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]

integrate(1/(sqrt(a + b/x^4)*x^2), x)