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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0199551, antiderivative size = 28, 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 \[ -\frac{2 x^{-n-1}}{b (3 n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(b*x^n)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.69021, size = 34, normalized size = 1.21 \[ - \frac{2 x^{- \frac{n}{2}} x^{- \frac{3 n}{2} - 1} \sqrt{b x^{n}}}{b^{2} \left (3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(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))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0199225, size = 22, normalized size = 0.79 \[ \frac{1}{\left (-\frac{3 n}{2}-1\right ) x \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

1/((-1 - (3*n)/2)*x*(b*x^n)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[ -2\,{\frac{1}{x \left ( 2+3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^n)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

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/((b*x^n)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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/((b*x^n)^(3/2)*x^2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n)^(3/2)*x^2), x)