3.1879 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{3 x}{8 a^2 \left (a x^2+b\right )}-\frac{x^3}{4 a \left (a x^2+b\right )^2} \]

[Out]

-x^3/(4*a*(b + a*x^2)^2) - (3*x)/(8*a^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqr
t[b]])/(8*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.0723402, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{3 x}{8 a^2 \left (a x^2+b\right )}-\frac{x^3}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^2),x]

[Out]

-x^3/(4*a*(b + a*x^2)^2) - (3*x)/(8*a^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqr
t[b]])/(8*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.5358, size = 56, normalized size = 0.88 \[ - \frac{x^{3}}{4 a \left (a x^{2} + b\right )^{2}} - \frac{3 x}{8 a^{2} \left (a x^{2} + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**2,x)

[Out]

-x**3/(4*a*(a*x**2 + b)**2) - 3*x/(8*a**2*(a*x**2 + b)) + 3*atan(sqrt(a)*x/sqrt(
b))/(8*a**(5/2)*sqrt(b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0909158, size = 55, normalized size = 0.86 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{5/2} \sqrt{b}}-\frac{5 a x^3+3 b x}{8 a^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^2),x]

[Out]

-(3*b*x + 5*a*x^3)/(8*a^2*(b + a*x^2)^2) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*a^
(5/2)*Sqrt[b])

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 47, normalized size = 0.7 \[{\frac{1}{ \left ( a{x}^{2}+b \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8\,a}}-{\frac{3\,bx}{8\,{a}^{2}}} \right ) }+{\frac{3}{8\,{a}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^2,x)

[Out]

(-5/8*x^3/a-3/8*b*x/a^2)/(a*x^2+b)^2+3/8/a^2/(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)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238947, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (\frac{2 \, a b x +{\left (a x^{2} - b\right )} \sqrt{-a b}}{a x^{2} + b}\right ) - 2 \,{\left (5 \, a x^{3} + 3 \, b x\right )} \sqrt{-a b}}{16 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{b}\right ) -{\left (5 \, a x^{3} + 3 \, b x\right )} \sqrt{a b}}{8 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^2),x, algorithm="fricas")

[Out]

[1/16*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*log((2*a*b*x + (a*x^2 - b)*sqrt(-a*b))/(a*x
^2 + b)) - 2*(5*a*x^3 + 3*b*x)*sqrt(-a*b))/((a^4*x^4 + 2*a^3*b*x^2 + a^2*b^2)*sq
rt(-a*b)), 1/8*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*arctan(sqrt(a*b)*x/b) - (5*a*x^3 +
3*b*x)*sqrt(a*b))/((a^4*x^4 + 2*a^3*b*x^2 + a^2*b^2)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 1.93344, size = 109, normalized size = 1.7 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (- a^{2} b \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} b}} \log{\left (a^{2} b \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{16} - \frac{5 a x^{3} + 3 b x}{8 a^{4} x^{4} + 16 a^{3} b x^{2} + 8 a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**2,x)

[Out]

-3*sqrt(-1/(a**5*b))*log(-a**2*b*sqrt(-1/(a**5*b)) + x)/16 + 3*sqrt(-1/(a**5*b))
*log(a**2*b*sqrt(-1/(a**5*b)) + x)/16 - (5*a*x**3 + 3*b*x)/(8*a**4*x**4 + 16*a**
3*b*x**2 + 8*a**2*b**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231845, size = 61, normalized size = 0.95 \[ \frac{3 \, \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2}} - \frac{5 \, a x^{3} + 3 \, b x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^2),x, algorithm="giac")

[Out]

3/8*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/8*(5*a*x^3 + 3*b*x)/((a*x^2 + b)^2
*a^2)