3.1895 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^3} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

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Rubi [A]  time = 0.029328, antiderivative size = 18, 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 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]/x^3,x]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

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Rubi in Sympy [A]  time = 2.12348, size = 14, normalized size = 0.78 \[ - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b/x**2)**(3/2)/(3*b)

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Mathematica [A]  time = 0.03088, size = 18, normalized size = 1. \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]/x^3,x]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

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Maple [A]  time = 0.007, size = 29, normalized size = 1.6 \[ -{\frac{a{x}^{2}+b}{3\,b{x}^{2}}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.57656, size = 19, normalized size = 1.06 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(a + b/x^2)^(3/2)/b

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Fricas [A]  time = 0.233625, size = 38, normalized size = 2.11 \[ -\frac{{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.49636, size = 42, normalized size = 2.33 \[ - \frac{a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{3 b} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.243901, size = 85, normalized size = 4.72 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}}{\rm sign}\left (x\right ) + a^{\frac{3}{2}} b^{2}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(3/2)*sign(x) + a^(3/2)*b^2*sign(x))/((
sqrt(a)*x - sqrt(a*x^2 + b))^2 - b)^3