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

Optimal. Leaf size=19 \[ \frac{x^3}{4 b \sqrt{\frac{b}{x^2}}} \]

[Out]

x^3/(4*b*Sqrt[b/x^2])

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Rubi [A]  time = 0.00855059, 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 \[ \frac{x^3}{4 b \sqrt{\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(b/x^2)^(-3/2),x]

[Out]

x^3/(4*b*Sqrt[b/x^2])

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Rubi in Sympy [A]  time = 1.55625, size = 15, normalized size = 0.79 \[ \frac{x^{5} \sqrt{\frac{b}{x^{2}}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*sqrt(b/x**2)/(4*b**2)

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Mathematica [A]  time = 0.00366509, size = 14, normalized size = 0.74 \[ \frac{x}{4 \left (\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/x^2)^(-3/2),x]

[Out]

x/(4*(b/x^2)^(3/2))

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Maple [A]  time = 0.003, size = 11, normalized size = 0.6 \[{\frac{x}{4} \left ({\frac{b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/x^2)^(3/2),x)

[Out]

1/4*x/(b/x^2)^(3/2)

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Maxima [A]  time = 1.52109, size = 14, normalized size = 0.74 \[ \frac{x}{4 \, \left (\frac{b}{x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x/(b/x^2)^(3/2)

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Fricas [A]  time = 0.222122, size = 20, normalized size = 1.05 \[ \frac{x^{5} \sqrt{\frac{b}{x^{2}}}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^5*sqrt(b/x^2)/b^2

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Sympy [A]  time = 1.88348, size = 15, normalized size = 0.79 \[ \frac{x}{4 b^{\frac{3}{2}} \left (\frac{1}{x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/x**2)**(3/2),x)

[Out]

x/(4*b**(3/2)*(x**(-2))**(3/2))

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GIAC/XCAS [A]  time = 0.222346, size = 11, normalized size = 0.58 \[ \frac{x^{4}}{4 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4/b^(3/2)