3.60 \(\int \frac{1}{\left (b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{4 b^2 x^3 \sqrt{b x^2}} \]

[Out]

-1/(4*b^2*x^3*Sqrt[b*x^2])

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Rubi [A]  time = 0.00827732, 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 \[ -\frac{1}{4 b^2 x^3 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2)^(-5/2),x]

[Out]

-1/(4*b^2*x^3*Sqrt[b*x^2])

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Rubi in Sympy [A]  time = 1.47175, size = 17, normalized size = 0.89 \[ - \frac{\sqrt{b x^{2}}}{4 b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2)**(5/2),x)

[Out]

-sqrt(b*x**2)/(4*b**3*x**5)

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Mathematica [A]  time = 0.0003683, size = 14, normalized size = 0.74 \[ -\frac{x}{4 \left (b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2)^(-5/2),x]

[Out]

-x/(4*(b*x^2)^(5/2))

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Maple [A]  time = 0.004, size = 11, normalized size = 0.6 \[ -{\frac{x}{4} \left ( b{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.43807, size = 11, normalized size = 0.58 \[ -\frac{1}{4 \, b^{\frac{5}{2}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(-5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224721, size = 20, normalized size = 1.05 \[ -\frac{\sqrt{b x^{2}}}{4 \, b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(-5/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(b*x^2)/(b^3*x^5)

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Sympy [A]  time = 3.06762, size = 15, normalized size = 0.79 \[ - \frac{x}{4 b^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.544794, size = 4, normalized size = 0.21 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2)^(-5/2),x, algorithm="giac")

[Out]

sage0*x