3.1866 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0385874, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^4),x]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.24233, size = 36, normalized size = 0.8 \[ \frac{x}{2 b \left (a x^{2} + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 \sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**2/x**4,x)

[Out]

x/(2*b*(a*x**2 + b)) + atan(sqrt(a)*x/sqrt(b))/(2*sqrt(a)*b**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0448949, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{x}{2 b \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^2*x^4),x]

[Out]

x/(2*b*(b + a*x^2)) + ArcTan[(Sqrt[a]*x)/Sqrt[b]]/(2*Sqrt[a]*b^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \[{\frac{x}{2\,b \left ( a{x}^{2}+b \right ) }}+{\frac{1}{2\,b}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^4,x)

[Out]

1/2*x/b/(a*x^2+b)+1/2/b/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.232135, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (a x^{2} + b\right )} \log \left (\frac{2 \, a b x +{\left (a x^{2} - b\right )} \sqrt{-a b}}{a x^{2} + b}\right ) + 2 \, \sqrt{-a b} x}{4 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{-a b}}, \frac{{\left (a x^{2} + b\right )} \arctan \left (\frac{\sqrt{a b} x}{b}\right ) + \sqrt{a b} x}{2 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^4),x, algorithm="fricas")

[Out]

[1/4*((a*x^2 + b)*log((2*a*b*x + (a*x^2 - b)*sqrt(-a*b))/(a*x^2 + b)) + 2*sqrt(-
a*b)*x)/((a*b*x^2 + b^2)*sqrt(-a*b)), 1/2*((a*x^2 + b)*arctan(sqrt(a*b)*x/b) + s
qrt(a*b)*x)/((a*b*x^2 + b^2)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 1.45556, size = 78, normalized size = 1.73 \[ \frac{x}{2 a b x^{2} + 2 b^{2}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (b^{2} \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**2/x**4,x)

[Out]

x/(2*a*b*x**2 + 2*b**2) - sqrt(-1/(a*b**3))*log(-b**2*sqrt(-1/(a*b**3)) + x)/4 +
sqrt(-1/(a*b**3))*log(b**2*sqrt(-1/(a*b**3)) + x)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232761, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b} + \frac{x}{2 \,{\left (a x^{2} + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^4),x, algorithm="giac")

[Out]

1/2*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b) + 1/2*x/((a*x^2 + b)*b)