3.1.98 \(\int \frac {1}{(\frac {b}{x})^{3/2}} \, dx\) [98]

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} \begin {gather*} \frac {2 x^2}{5 b \sqrt {\frac {b}{x}}} \end {gather*}

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 \begin {gather*} \frac {2 x}{5 \left (\frac {b}{x}\right )^{3/2}} \end {gather*}

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 = 0.02, size = 11, normalized size = 0.58

method	result	size
gosper	\(\frac {2 x}{5 \left (\frac {b}{x}\right )^{\frac {3}{2}}}\)	\(11\)
default	\(\frac {2 x}{5 \left (\frac {b}{x}\right )^{\frac {3}{2}}}\)	\(11\)
trager	\(\frac {2 x^{3} \sqrt {\frac {b}{x}}}{5 b^{2}}\)	\(16\)
risch	\(\frac {2 x^{2}}{5 b \sqrt {\frac {b}{x}}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 10, normalized size = 0.53 \begin {gather*} \frac {2 \, x}{5 \, \left (\frac {b}{x}\right )^{\frac {3}{2}}} \end {gather*}

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 = 0.38, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 \, x^{3} \sqrt {\frac {b}{x}}}{5 \, b^{2}} \end {gather*}

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 = 0.19, size = 10, normalized size = 0.53 \begin {gather*} \frac {2 x}{5 \left (\frac {b}{x}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.73, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {b x} x^{2}}{5 \, b^{2} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.94, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,x^3\,\sqrt {\frac {b}{x}}}{5\,b^2} \end {gather*}

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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