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

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

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Rubi [A]  time = 0.00, antiderivative size = 12, 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} \begin {gather*} -2 b \sqrt {\frac {b}{x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -2 x \left (\frac {b}{x}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -2 b \sqrt {\frac {b}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b*Sqrt[b/x]

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fricas [A]  time = 0.76, size = 10, normalized size = 0.83 \begin {gather*} -2 \, b \sqrt {\frac {b}{x}} \end {gather*}

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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giac [A]  time = 0.16, size = 12, normalized size = 1.00 \begin {gather*} -\frac {2 \, b^{2} \mathrm {sgn}\relax (x)}{\sqrt {b x}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} -2 \left (\frac {b}{x}\right )^{\frac {3}{2}} x \end {gather*}

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 = 1.32, size = 10, normalized size = 0.83 \begin {gather*} -2 \, x \left (\frac {b}{x}\right )^{\frac {3}{2}} \end {gather*}

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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mupad [B]  time = 0.93, size = 10, normalized size = 0.83 \begin {gather*} -2\,b\,\sqrt {\frac {b}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.42, size = 15, normalized size = 1.25 \begin {gather*} - 2 b^{\frac {3}{2}} x \left (\frac {1}{x}\right )^{\frac {3}{2}} \end {gather*}

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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