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

Optimal. Leaf size=19 \[ \frac{2 x^2}{5 b \sqrt{\frac{b}{x}}} \]

[Out]

(2*x^2)/(5*b*Sqrt[b/x])

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Rubi [A]  time = 0.00865234, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 x^2}{5 b \sqrt{\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(b/x)^(-3/2),x]

[Out]

(2*x^2)/(5*b*Sqrt[b/x])

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Rubi in Sympy [A]  time = 1.61813, size = 15, normalized size = 0.79 \[ \frac{2 x^{3} \sqrt{\frac{b}{x}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**3*sqrt(b/x)/(5*b**2)

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Mathematica [A]  time = 0.00404491, size = 14, normalized size = 0.74 \[ \frac{2 x}{5 \left (\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x)^(-3/2),x]

[Out]

(2*x)/(5*(b/x)^(3/2))

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Maple [A]  time = 0.003, size = 11, normalized size = 0.6 \[{\frac{2\,x}{5} \left ({\frac{b}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x)^(3/2),x)

[Out]

2/5*x/(b/x)^(3/2)

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Maxima [A]  time = 1.45064, size = 14, normalized size = 0.74 \[ \frac{2 \, x}{5 \, \left (\frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x/(b/x)^(3/2)

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Fricas [A]  time = 0.22845, size = 20, normalized size = 1.05 \[ \frac{2 \, x^{2}}{5 \, b \sqrt{\frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x^2/(b*sqrt(b/x))

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Sympy [A]  time = 1.85114, size = 15, normalized size = 0.79 \[ \frac{2 x}{5 b^{\frac{3}{2}} \left (\frac{1}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x)**(3/2),x)

[Out]

2*x/(5*b**(3/2)*(1/x)**(3/2))

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GIAC/XCAS [A]  time = 0.235003, size = 20, normalized size = 1.05 \[ \frac{2 \, x^{2}}{5 \, b \sqrt{\frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x^2/(b*sqrt(b/x))