3.1788 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{3/2}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

[Out]

-2/(a*Sqrt[a + b/x]*Sqrt[x])

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Rubi [A]  time = 0.025686, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2}{a \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(a*Sqrt[a + b/x]*Sqrt[x])

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Rubi in Sympy [A]  time = 2.73657, size = 17, normalized size = 0.81 \[ - \frac{2}{a \sqrt{x} \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*sqrt(x)*sqrt(a + b/x))

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Mathematica [A]  time = 0.0339656, size = 30, normalized size = 1.43 \[ -\frac{2 \sqrt{x} \sqrt{\frac{a x+b}{x}}}{a (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x]*Sqrt[(b + a*x)/x])/(a*(b + a*x))

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Maple [A]  time = 0.004, size = 25, normalized size = 1.2 \[ -2\,{\frac{ax+b}{a{x}^{3/2}} \left ({\frac{ax+b}{x}} \right ) ^{-3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43356, size = 23, normalized size = 1.1 \[ -\frac{2}{\sqrt{a + \frac{b}{x}} a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(a + b/x)*a*sqrt(x))

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Fricas [A]  time = 0.234193, size = 26, normalized size = 1.24 \[ -\frac{2}{a \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*sqrt(x)*sqrt((a*x + b)/x))

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Sympy [A]  time = 107.435, size = 19, normalized size = 0.9 \[ - \frac{2}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*sqrt(b)*sqrt(a*x/b + 1))

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GIAC/XCAS [A]  time = 0.232329, size = 28, normalized size = 1.33 \[ -\frac{2}{\sqrt{a x + b} a} + \frac{2}{a \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(a*x + b)*a) + 2/(a*sqrt(b))