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

Optimal. Leaf size=17 \[ -\frac{2 b \sqrt{\frac{b}{x^3}}}{7 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0017197, antiderivative size = 17, 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} \[ -\frac{2 b \sqrt{\frac{b}{x^3}}}{7 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0012279, size = 14, normalized size = 0.82 \[ -\frac{2}{7} x \left (\frac{b}{x^3}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.7 \begin{align*} -{\frac{2\,x}{7} \left ({\frac{b}{{x}^{3}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995202, size = 14, normalized size = 0.82 \begin{align*} -\frac{2}{7} \, x \left (\frac{b}{x^{3}}\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71544, size = 32, normalized size = 1.88 \begin{align*} -\frac{2 \, b \sqrt{\frac{b}{x^{3}}}}{7 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.586658, size = 19, normalized size = 1.12 \begin{align*} - \frac{2 b^{\frac{3}{2}} x \left (\frac{1}{x^{3}}\right )^{\frac{3}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16894, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \, b^{2}}{7 \, \sqrt{b x} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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