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

Optimal. Leaf size=12 \[ -2 x \sqrt{\frac{b}{x^3}} \]

[Out]

-2*Sqrt[b/x^3]*x

________________________________________________________________________________________

Rubi [A]  time = 0.0016037, antiderivative size = 12, 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} \[ -2 x \sqrt{\frac{b}{x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[b/x^3]*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 \sqrt{\frac{b}{x^3}} \, dx &=\left (\sqrt{\frac{b}{x^3}} x^{3/2}\right ) \int \frac{1}{x^{3/2}} \, dx\\ &=-2 \sqrt{\frac{b}{x^3}} x\\ \end{align*}

Mathematica [A]  time = 0.0012782, size = 12, normalized size = 1. \[ -2 x \sqrt{\frac{b}{x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[b/x^3]*x

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} -2\,x\sqrt{{\frac{b}{{x}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.96852, size = 14, normalized size = 1.17 \begin{align*} -2 \, x \sqrt{\frac{b}{x^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.64766, size = 24, normalized size = 2. \begin{align*} -2 \, x \sqrt{\frac{b}{x^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.41267, size = 17, normalized size = 1.42 \begin{align*} - 2 \sqrt{b} x \sqrt{\frac{1}{x^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)*x*sqrt(x**(-3))

________________________________________________________________________________________

Giac [A]  time = 1.14144, size = 11, normalized size = 0.92 \begin{align*} -\frac{2 \, b}{\sqrt{b x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b/sqrt(b*x)