3.8 \(\int \frac{\sqrt{b x^2}}{x^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{\sqrt{b x^2}}{x^2} \]

[Out]

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

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Rubi [A]  time = 0.0015441, antiderivative size = 14, 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} \[ -\frac{\sqrt{b x^2}}{x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2]/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0015793, size = 14, normalized size = 1. \[ -\frac{\sqrt{b x^2}}{x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2]/x^3,x]

[Out]

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

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{{x}^{2}}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.24302, size = 24, normalized size = 1.71 \begin{align*} -\frac{\sqrt{b x^{2}}}{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.533543, size = 15, normalized size = 1.07 \begin{align*} - \frac{\sqrt{b} \sqrt{x^{2}}}{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17803, size = 14, normalized size = 1. \begin{align*} -\frac{\sqrt{b} \mathrm{sgn}\left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b)*sgn(x)/x