3.99 \(\int \frac {1}{(\frac {b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac {x^3}{4 b \sqrt {\frac {b}{x^2}}} \]

[Out]

1/4*x^3/b/(b/x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac {x^3}{4 b \sqrt {\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(b/x^2)^(-3/2),x]

[Out]

x^3/(4*b*Sqrt[b/x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (\frac {b}{x^2}\right )^{3/2}} \, dx &=\frac {\int x^3 \, dx}{b \sqrt {\frac {b}{x^2}} x}\\ &=\frac {x^3}{4 b \sqrt {\frac {b}{x^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {x}{4 \left (\frac {b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x^2)^(-3/2),x]

[Out]

x/(4*(b/x^2)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 15, normalized size = 0.79 \[ \frac {x^{5} \sqrt {\frac {b}{x^{2}}}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^2)^(3/2),x, algorithm="fricas")

[Out]

1/4*x^5*sqrt(b/x^2)/b^2

________________________________________________________________________________________

giac [A]  time = 0.16, size = 8, normalized size = 0.42 \[ \frac {x^{4}}{4 \, b^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^2)^(3/2),x, algorithm="giac")

[Out]

1/4*x^4/b^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 11, normalized size = 0.58 \[ \frac {x}{4 \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2)^(3/2),x)

[Out]

1/4*x/(b/x^2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 10, normalized size = 0.53 \[ \frac {x}{4 \, \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x^2)^(3/2),x, algorithm="maxima")

[Out]

1/4*x/(b/x^2)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.93, size = 13, normalized size = 0.68 \[ \frac {x^5\,\sqrt {\frac {1}{x^2}}}{4\,b^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2)^(3/2),x)

[Out]

(x^5*(1/x^2)^(1/2))/(4*b^(3/2))

________________________________________________________________________________________

sympy [A]  time = 0.51, size = 15, normalized size = 0.79 \[ \frac {x}{4 b^{\frac {3}{2}} \left (\frac {1}{x^{2}}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x**2)**(3/2),x)

[Out]

x/(4*b**(3/2)*(x**(-2))**(3/2))

________________________________________________________________________________________