3.1725 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^2} \, dx\)

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

[Out]

(-2*Sqrt[a + b/x])/b

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Rubi [A]  time = 0.0262674, 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 \[ -\frac{2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^2),x]

[Out]

(-2*Sqrt[a + b/x])/b

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Rubi in Sympy [A]  time = 2.21049, size = 12, normalized size = 0.75 \[ - \frac{2 \sqrt{a + \frac{b}{x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a + b/x)/b

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Mathematica [A]  time = 0.0136972, size = 16, normalized size = 1. \[ -\frac{2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b/x])/b

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Maple [A]  time = 0.007, size = 25, normalized size = 1.6 \[ -2\,{\frac{ax+b}{bx}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b/x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44046, 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/(sqrt(a + b/x)*x^2),x, algorithm="maxima")

[Out]

-2*sqrt(a + b/x)/b

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Fricas [A]  time = 0.224532, size = 22, normalized size = 1.38 \[ -\frac{2 \, \sqrt{\frac{a x + b}{x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt((a*x + b)/x)/b

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Sympy [A]  time = 2.87512, size = 22, normalized size = 1.38 \[ \begin{cases} - \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{for}\: b \neq 0 \\- \frac{1}{\sqrt{a} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.230186, 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/(sqrt(a + b/x)*x^2),x, algorithm="giac")

[Out]

-2*sqrt(a + b/x)/b