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

Optimal. Leaf size=19 \[ -\frac{1}{2 b x \sqrt{b x^2}} \]

[Out]

-1/(2*b*x*Sqrt[b*x^2])

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Rubi [A]  time = 0.00183, 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} \[ -\frac{1}{2 b x \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0011816, size = 14, normalized size = 0.74 \[ -\frac{x}{2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 11, normalized size = 0.6 \begin{align*} -{\frac{x}{2} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961178, size = 11, normalized size = 0.58 \begin{align*} -\frac{1}{2 \, b^{\frac{3}{2}} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5847, size = 38, normalized size = 2. \begin{align*} -\frac{\sqrt{b x^{2}}}{2 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.686693, size = 15, normalized size = 0.79 \begin{align*} - \frac{x}{2 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x