Optimal. Leaf size=55 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{5/2}}+\frac{3 x}{2 a^2}-\frac{x^3}{2 a \left (a x^2+b\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0608883, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{5/2}}+\frac{3 x}{2 a^2}-\frac{x^3}{2 a \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(-2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.2996, size = 48, normalized size = 0.87 \[ - \frac{x^{3}}{2 a \left (a x^{2} + b\right )} + \frac{3 x}{2 a^{2}} - \frac{3 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0577473, size = 51, normalized size = 0.93 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{5/2}}+\frac{b x}{2 a^2 \left (a x^2+b\right )}+\frac{x}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(-2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 43, normalized size = 0.8 \[{\frac{x}{{a}^{2}}}+{\frac{bx}{2\,{a}^{2} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,b}{2\,{a}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^2,x)
[Out]
_______________________________________________________________________________________
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((a + b/x^2)^(-2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.233937, size = 1, normalized size = 0.02 \[ \left [\frac{4 \, a x^{3} + 3 \,{\left (a x^{2} + b\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 6 \, b x}{4 \,{\left (a^{3} x^{2} + a^{2} b\right )}}, \frac{2 \, a x^{3} - 3 \,{\left (a x^{2} + b\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right ) + 3 \, b x}{2 \,{\left (a^{3} x^{2} + a^{2} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.59491, size = 83, normalized size = 1.51 \[ \frac{b x}{2 a^{3} x^{2} + 2 a^{2} b} + \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (- a^{2} \sqrt{- \frac{b}{a^{5}}} + x \right )}}{4} - \frac{3 \sqrt{- \frac{b}{a^{5}}} \log{\left (a^{2} \sqrt{- \frac{b}{a^{5}}} + x \right )}}{4} + \frac{x}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.221431, size = 57, normalized size = 1.04 \[ -\frac{3 \, b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} + \frac{x}{a^{2}} + \frac{b x}{2 \,{\left (a x^{2} + b\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(-2),x, algorithm="giac")
[Out]