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]

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

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Rubi [A]  time = 0.0020934, 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, 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 = 0.0016903, 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))

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Maple [A]  time = 0.001, size = 11, normalized size = 0.6 \begin{align*}{\frac{x}{4} \left ({\frac{b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.9582, size = 14, normalized size = 0.74 \begin{align*} \frac{x}{4 \, \left (\frac{b}{x^{2}}\right )^{\frac{3}{2}}} \end{align*}

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)

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Fricas [A]  time = 1.68737, size = 34, normalized size = 1.79 \begin{align*} \frac{x^{5} \sqrt{\frac{b}{x^{2}}}}{4 \, b^{2}} \end{align*}

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

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Sympy [A]  time = 0.656018, size = 15, normalized size = 0.79 \begin{align*} \frac{x}{4 b^{\frac{3}{2}} \left (\frac{1}{x^{2}}\right )^{\frac{3}{2}}} \end{align*}

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))

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Giac [A]  time = 1.17682, size = 11, normalized size = 0.58 \begin{align*} \frac{x^{4}}{4 \, b^{\frac{3}{2}}} \end{align*}

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)