3.151 \(\int \frac{1}{\left (b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{2 x^{1-n}}{b (2-3 n) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0201196, antiderivative size = 28, 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 \[ \frac{2 x^{1-n}}{b (2-3 n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.88221, size = 31, normalized size = 1.11 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} + 1} \sqrt{b x^{n}}}{b^{2} \left (- 3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**n)**(3/2),x)

[Out]

2*x**(-n/2)*x**(-3*n/2 + 1)*sqrt(b*x**n)/(b**2*(-3*n + 2))

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Mathematica [A]  time = 0.0117402, size = 20, normalized size = 0.71 \[ \frac{x}{\left (1-\frac{3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(-3/2),x]

[Out]

x/((1 - (3*n)/2)*(b*x^n)^(3/2))

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Maple [A]  time = 0.001, size = 18, normalized size = 0.6 \[ -2\,{\frac{x}{ \left ( -2+3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^n)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^(-3/2),x, algorithm="giac")

[Out]

integrate((b*x^n)^(-3/2), x)