3.1862 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2} \, dx\)

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]

(3*x)/(2*a^2) - x^3/(2*a*(b + a*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/
(2*a^(5/2))

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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]

(3*x)/(2*a^2) - x^3/(2*a*(b + a*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/
(2*a^(5/2))

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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]

-x**3/(2*a*(a*x**2 + b)) + 3*x/(2*a**2) - 3*sqrt(b)*atan(sqrt(a)*x/sqrt(b))/(2*a
**(5/2))

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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]

x/a^2 + (b*x)/(2*a^2*(b + a*x^2)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(2*a
^(5/2))

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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]

x/a^2+1/2*b/a^2*x/(a*x^2+b)-3/2*b/a^2/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

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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]

Exception raised: ValueError

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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]

[1/4*(4*a*x^3 + 3*(a*x^2 + b)*sqrt(-b/a)*log((a*x^2 - 2*a*x*sqrt(-b/a) - b)/(a*x
^2 + b)) + 6*b*x)/(a^3*x^2 + a^2*b), 1/2*(2*a*x^3 - 3*(a*x^2 + b)*sqrt(b/a)*arct
an(x/sqrt(b/a)) + 3*b*x)/(a^3*x^2 + a^2*b)]

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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]

b*x/(2*a**3*x**2 + 2*a**2*b) + 3*sqrt(-b/a**5)*log(-a**2*sqrt(-b/a**5) + x)/4 -
3*sqrt(-b/a**5)*log(a**2*sqrt(-b/a**5) + x)/4 + x/a**2

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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]

-3/2*b*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^2) + x/a^2 + 1/2*b*x/((a*x^2 + b)*a^2)