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

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

[Out]

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

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Rubi [A]  time = 0.0014569, 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} \[ -\frac{2 x}{\sqrt{b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0011769, size = 12, normalized size = 1. \[ -\frac{2 x}{\sqrt{b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 11, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{\sqrt{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*x/(b*x^3)^(1/2)

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Maxima [A]  time = 0.965119, size = 14, normalized size = 1.17 \begin{align*} -\frac{2 \, x}{\sqrt{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*x/sqrt(b*x^3)

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Fricas [A]  time = 1.69723, size = 32, normalized size = 2.67 \begin{align*} -\frac{2 \, \sqrt{b x^{3}}}{b x^{2}} \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*sqrt(b*x^3)/(b*x^2)

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Sympy [A]  time = 0.523199, size = 15, normalized size = 1.25 \begin{align*} - \frac{2 x}{\sqrt{b} \sqrt{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/(sqrt(b)*sqrt(x**3))

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Giac [A]  time = 1.11946, size = 15, normalized size = 1.25 \begin{align*} -\frac{2}{\sqrt{b x} \mathrm{sgn}\left (x\right )} \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/(sqrt(b*x)*sgn(x))