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]

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

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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} \[ \frac {2 x^2}{5 b \sqrt {\frac {b}{x}}} \]

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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.92, size = 15, normalized size = 0.79 \[ \frac {2 \, x^{3} \sqrt {\frac {b}{x}}}{5 \, b^{2}} \]

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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giac [A]  time = 0.17, size = 15, normalized size = 0.79 \[ \frac {2 \, x^{2}}{5 \, b \sqrt {\frac {b}{x}}} \]

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))

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maple [A]  time = 0.00, size = 11, normalized size = 0.58 \[ \frac {2 x}{5 \left (\frac {b}{x}\right )^{\frac {3}{2}}} \]

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 = 1.32, size = 10, normalized size = 0.53 \[ \frac {2 \, x}{5 \, \left (\frac {b}{x}\right )^{\frac {3}{2}}} \]

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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mupad [B]  time = 0.94, size = 15, normalized size = 0.79 \[ \frac {2\,x^3\,\sqrt {\frac {b}{x}}}{5\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.50, size = 15, normalized size = 0.79 \[ \frac {2 x}{5 b^{\frac {3}{2}} \left (\frac {1}{x}\right )^{\frac {3}{2}}} \]

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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