3.95 \(\int \frac{1}{(b x^3)^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2}{7 b x^2 \sqrt{b x^3}} \]

[Out]

-2/(7*b*x^2*Sqrt[b*x^3])

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Rubi [A]  time = 0.0017849, antiderivative size = 19, 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} \[ -\frac{2}{7 b x^2 \sqrt{b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(7*b*x^2*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}{\left (b x^3\right )^{3/2}} \, dx &=\frac{x^{3/2} \int \frac{1}{x^{9/2}} \, dx}{b \sqrt{b x^3}}\\ &=-\frac{2}{7 b x^2 \sqrt{b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0016715, size = 14, normalized size = 0.74 \[ -\frac{2 x}{7 \left (b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(7*(b*x^3)^(3/2))

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Maple [A]  time = 0.003, size = 11, normalized size = 0.6 \begin{align*} -{\frac{2\,x}{7} \left ( b{x}^{3} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*x/(b*x^3)^(3/2)

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Maxima [A]  time = 0.988873, size = 14, normalized size = 0.74 \begin{align*} -\frac{2 \, x}{7 \, \left (b x^{3}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*x/(b*x^3)^(3/2)

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Fricas [A]  time = 1.63251, size = 38, normalized size = 2. \begin{align*} -\frac{2 \, \sqrt{b x^{3}}}{7 \, b^{2} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*sqrt(b*x^3)/(b^2*x^5)

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Sympy [A]  time = 0.507006, size = 17, normalized size = 0.89 \begin{align*} - \frac{2 x}{7 b^{\frac{3}{2}} \left (x^{3}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/(7*b**(3/2)*(x**3)**(3/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x