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

Optimal. Leaf size=12 \[ -2 b \sqrt{\frac{b}{x}} \]

[Out]

-2*b*Sqrt[b/x]

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Rubi [A]  time = 0.0016709, antiderivative size = 12, 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} \[ -2 b \sqrt{\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0011895, size = 12, normalized size = 1. \[ -2 x \left (\frac{b}{x}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} -2\,x \left ({\frac{b}{x}} \right ) ^{3/2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.958135, size = 14, normalized size = 1.17 \begin{align*} -2 \, x \left (\frac{b}{x}\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.72821, size = 22, normalized size = 1.83 \begin{align*} -2 \, b \sqrt{\frac{b}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*sqrt(b/x)

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Sympy [A]  time = 0.504955, size = 15, normalized size = 1.25 \begin{align*} - 2 b^{\frac{3}{2}} x \left (\frac{1}{x}\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12657, size = 16, normalized size = 1.33 \begin{align*} -\frac{2 \, b^{2} \mathrm{sgn}\left (x\right )}{\sqrt{b x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^2*sgn(x)/sqrt(b*x)