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

Optimal. Leaf size=21 \[ -\frac{2 \sqrt{b x^n}}{(2-n) x} \]

[Out]

(-2*Sqrt[b*x^n])/((2 - n)*x)

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Rubi [A]  time = 0.0153905, antiderivative size = 21, 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 \[ -\frac{2 \sqrt{b x^n}}{(2-n) x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^n]/x^2,x]

[Out]

(-2*Sqrt[b*x^n])/((2 - n)*x)

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Rubi in Sympy [A]  time = 2.9047, size = 26, normalized size = 1.24 \[ - \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} - 1} \sqrt{b x^{n}}}{- n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00386763, size = 19, normalized size = 0.9 \[ \frac{2 \sqrt{b x^n}}{(n-2) x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^n]/x^2,x]

[Out]

(2*Sqrt[b*x^n])/((-2 + n)*x)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \[ 2\,{\frac{\sqrt{b{x}^{n}}}{x \left ( -2+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(1/2)/x^2,x)

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n)/x^2, x)