3.94 \(\int \frac{1}{\sqrt{\frac{b}{x^3}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0014239, antiderivative size = 14, 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} \[ \frac{2 x}{5 \sqrt{\frac{b}{x^3}}} \]

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*}

Mathematica [A]  time = 0.0016246, size = 14, normalized size = 1. \[ \frac{2 x}{5 \sqrt{\frac{b}{x^3}}} \]

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 = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{2\,x}{5}{\frac{1}{\sqrt{{\frac{b}{{x}^{3}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955916, size = 14, normalized size = 1. \begin{align*} \frac{2 \, x}{5 \, \sqrt{\frac{b}{x^{3}}}} \end{align*}

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 = 1.69002, size = 31, normalized size = 2.21 \begin{align*} \frac{2 \, x^{4} \sqrt{\frac{b}{x^{3}}}}{5 \, b} \end{align*}

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 = 0.39797, size = 17, normalized size = 1.21 \begin{align*} \frac{2 x}{5 \sqrt{b} \sqrt{\frac{1}{x^{3}}}} \end{align*}

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)*sqrt(x**(-3)))

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Giac [A]  time = 1.17394, size = 18, normalized size = 1.29 \begin{align*} \frac{2 \, \sqrt{b x} x^{2}}{5 \, b} \end{align*}

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