3.1.94 \(\int \frac {1}{\sqrt {\frac {b}{x^3}}} \, dx\) [94]

Optimal. Leaf size=14 \[ \frac {2 x}{5 \sqrt {\frac {b}{x^3}}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, 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}{5 \sqrt {\frac {b}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b/x^3],x]

[Out]

(2*x)/(5*Sqrt[b/x^3])

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

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 x}{5 \sqrt {\frac {b}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b/x^3],x]

[Out]

(2*x)/(5*Sqrt[b/x^3])

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Maple [A]
time = 0.02, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {2 x}{5 \sqrt {\frac {b}{x^{3}}}}\)	\(11\)
default	\(\frac {2 x}{5 \sqrt {\frac {b}{x^{3}}}}\)	\(11\)
risch	\(\frac {2 x}{5 \sqrt {\frac {b}{x^{3}}}}\)	\(11\)
trager	\(\frac {2 \sqrt {\frac {b}{x^{3}}}\, x^{4}}{5 b}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \, x}{5 \, \sqrt {\frac {b}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 15, normalized size = 1.07 \begin {gather*} \frac {2 \, x^{4} \sqrt {\frac {b}{x^{3}}}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^4*sqrt(b/x^3)/b

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Sympy [A]
time = 0.17, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 x}{5 \sqrt {\frac {b}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.81, size = 13, normalized size = 0.93 \begin {gather*} \frac {2 \, \sqrt {b x} x^{2}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.95, size = 15, normalized size = 1.07 \begin {gather*} \frac {2\,x^4\,\sqrt {\frac {b}{x^3}}}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^4*(b/x^3)^(1/2))/(5*b)

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