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

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

[Out]

(2*x)/((2 - n)*Sqrt[b*x^n])

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Rubi [A]  time = 0.0126432, 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 \[ \frac{2 x}{(2-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[b*x^n],x]

[Out]

(2*x)/((2 - n)*Sqrt[b*x^n])

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Rubi in Sympy [A]  time = 2.62409, size = 26, normalized size = 1.37 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{n}{2} + 1} \sqrt{b x^{n}}}{b \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(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 = 0.00474119, size = 17, normalized size = 0.89 \[ -\frac{2 x}{(n-2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[b*x^n],x]

[Out]

(-2*x)/((-2 + n)*Sqrt[b*x^n])

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Maple [A]  time = 0.001, size = 16, normalized size = 0.8 \[ -2\,{\frac{x}{ \left ( -2+n \right ) \sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^n)^(1/2),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(1/sqrt(b*x^n),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, 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/(b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(b*x^n), x)