3.1727 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^2} \, dx\)

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

[Out]

(-2*Sqrt[a + b/x])/b

________________________________________________________________________________________

Rubi [A]  time = 0.0051254, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x^2),x]

[Out]

(-2*Sqrt[a + b/x])/b

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} x^2} \, dx &=-\frac{2 \sqrt{a+\frac{b}{x}}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0059379, size = 16, normalized size = 1. \[ -\frac{2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^2),x]

[Out]

(-2*Sqrt[a + b/x])/b

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 25, normalized size = 1.6 \begin{align*} -2\,{\frac{ax+b}{bx}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b/x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.18039, size = 19, normalized size = 1.19 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a + b/x)/b

________________________________________________________________________________________

Fricas [A]  time = 1.59524, size = 32, normalized size = 2. \begin{align*} -\frac{2 \, \sqrt{\frac{a x + b}{x}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt((a*x + b)/x)/b

________________________________________________________________________________________

Sympy [A]  time = 0.741817, size = 22, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{for}\: b \neq 0 \\- \frac{1}{\sqrt{a} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(a + b/x)/b, Ne(b, 0)), (-1/(sqrt(a)*x), True))

________________________________________________________________________________________

Giac [A]  time = 1.10221, size = 19, normalized size = 1.19 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

-2*sqrt(a + b/x)/b