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

Optimal. Leaf size=19 \[ -\frac{2}{(n+2) x \sqrt{b x^n}} \]

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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Rubi [A]  time = 0.0137036, antiderivative size = 19, 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 \[ -\frac{2}{(n+2) x \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[b*x^n]),x]

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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Rubi in Sympy [A]  time = 3.59663, size = 29, normalized size = 1.53 \[ - \frac{2 x^{- \frac{n}{2}} x^{- \frac{n}{2} - 1} \sqrt{b x^{n}}}{b \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00719642, size = 19, normalized size = 1. \[ -\frac{2}{(n+2) x \sqrt{b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \[ -2\,{\frac{1}{ \left ( 2+n \right ) x\sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x^n)*x^2), x)