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

Optimal. Leaf size=19 \[ \frac{2 x^2}{5 b \sqrt{\frac{b}{x}}} \]

[Out]

(2*x^2)/(5*b*Sqrt[b/x])

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Rubi [A]  time = 0.0020271, 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} \[ \frac{2 x^2}{5 b \sqrt{\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2)/(5*b*Sqrt[b/x])

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}\right )^{3/2}} \, dx &=\frac{\int x^{3/2} \, dx}{b \sqrt{\frac{b}{x}} \sqrt{x}}\\ &=\frac{2 x^2}{5 b \sqrt{\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0016948, size = 14, normalized size = 0.74 \[ \frac{2 x}{5 \left (\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.6 \begin{align*}{\frac{2\,x}{5} \left ({\frac{b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.98451, size = 14, normalized size = 0.74 \begin{align*} \frac{2 \, x}{5 \, \left (\frac{b}{x}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68892, size = 31, normalized size = 1.63 \begin{align*} \frac{2 \, x^{3} \sqrt{\frac{b}{x}}}{5 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^3*sqrt(b/x)/b^2

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Sympy [A]  time = 0.642703, size = 15, normalized size = 0.79 \begin{align*} \frac{2 x}{5 b^{\frac{3}{2}} \left (\frac{1}{x}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15236, size = 20, normalized size = 1.05 \begin{align*} \frac{2 \, x^{2}}{5 \, b \sqrt{\frac{b}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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