3.1734 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

[Out]

2/(b*Sqrt[a + b/x])

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Rubi [A]  time = 0.0271752, antiderivative size = 16, 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 \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^2),x]

[Out]

2/(b*Sqrt[a + b/x])

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Rubi in Sympy [A]  time = 2.17046, size = 10, normalized size = 0.62 \[ \frac{2}{b \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b*sqrt(a + b/x))

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Mathematica [A]  time = 0.0219579, size = 16, normalized size = 1. \[ \frac{2}{b \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

2/(b*Sqrt[a + b/x])

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Maple [A]  time = 0.008, size = 25, normalized size = 1.6 \[ 2\,{\frac{ax+b}{bx} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43695, size = 19, normalized size = 1.19 \[ \frac{2}{\sqrt{a + \frac{b}{x}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(sqrt(a + b/x)*b)

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Fricas [A]  time = 0.230679, size = 22, normalized size = 1.38 \[ \frac{2}{b \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b*sqrt((a*x + b)/x))

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Sympy [A]  time = 4.43499, size = 20, normalized size = 1.25 \[ \begin{cases} \frac{2}{b \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{a^{\frac{3}{2}} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2/(b*sqrt(a + b/x)), Ne(b, 0)), (-1/(a**(3/2)*x), True))

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GIAC/XCAS [A]  time = 0.263057, size = 19, normalized size = 1.19 \[ \frac{2}{\sqrt{a + \frac{b}{x}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(sqrt(a + b/x)*b)