3.41 \(\int \frac{1}{x^2 \sqrt{b x^2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{2 x \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.00664285, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{2 x \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[b*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.49238, size = 15, normalized size = 0.94 \[ - \frac{\sqrt{b x^{2}}}{2 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00420618, size = 15, normalized size = 0.94 \[ -\frac{b x}{2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \[ -{\frac{1}{2\,x}{\frac{1}{\sqrt{b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.41343, size = 11, normalized size = 0.69 \[ -\frac{1}{2 \, \sqrt{b} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(sqrt(b)*x^2)

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Fricas [A]  time = 0.21336, size = 20, normalized size = 1.25 \[ -\frac{\sqrt{b x^{2}}}{2 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.91366, size = 17, normalized size = 1.06 \[ - \frac{1}{2 \sqrt{b} x \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*sqrt(b)*x*sqrt(x**2))

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GIAC/XCAS [A]  time = 0.572773, size = 4, normalized size = 0.25 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x