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

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, 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}{7 b x^2 \sqrt {b x^3}} \end {gather*}

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 = 0.00, size = 14, normalized size = 0.74 \begin {gather*} -\frac {2 x}{7 \left (b x^3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.01, size = 14, normalized size = 0.74 \begin {gather*} -\frac {2 x}{7 \left (b x^3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2 \, \sqrt {b x^{3}}}{7 \, b^{2} x^{5}} \end {gather*}

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.00, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 x}{7 \left (b \,x^{3}\right )^{\frac {3}{2}}} \end {gather*}

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)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 10, normalized size = 0.53 \begin {gather*} -\frac {2 \, x}{7 \, \left (b x^{3}\right )^{\frac {3}{2}}} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.94, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2\,\sqrt {b\,x^3}}{7\,b^2\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(b*x^3)^(1/2))/(7*b^2*x^5)

________________________________________________________________________________________

sympy [A]  time = 0.51, size = 17, normalized size = 0.89 \begin {gather*} - \frac {2 x}{7 b^{\frac {3}{2}} \left (x^{3}\right )^{\frac {3}{2}}} \end {gather*}

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))

________________________________________________________________________________________