3.41 \(\int \frac{1}{x^2 \sqrt{b x^2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{2 x \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0014971, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac{1}{2 x \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0016589, size = 15, normalized size = 0.94 \[ -\frac{b x}{2 \left (b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,x}{\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.984722, size = 11, normalized size = 0.69 \begin{align*} -\frac{1}{2 \, \sqrt{b} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43818, size = 35, normalized size = 2.19 \begin{align*} -\frac{\sqrt{b x^{2}}}{2 \, b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.646963, size = 17, normalized size = 1.06 \begin{align*} - \frac{1}{2 \sqrt{b} x \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*sqrt(b)*x*sqrt(x**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/x^2/(b*x^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x