3.1938 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.0298115, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.79089, size = 15, normalized size = 0.79 \[ - \frac{1}{a x \sqrt{a + \frac{b}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*x*sqrt(a + b/x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.010665, size = 19, normalized size = 1. \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 29, normalized size = 1.5 \[ -{\frac{a{x}^{2}+b}{a{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4321, size = 23, normalized size = 1.21 \[ -\frac{1}{\sqrt{a + \frac{b}{x^{2}}} a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(sqrt(a + b/x^2)*a*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.230548, size = 39, normalized size = 2.05 \[ -\frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{2} x^{2} + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.57052, size = 20, normalized size = 1.05 \[ - \frac{1}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*sqrt(b)*sqrt(a*x**2/b + 1))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^2), x)