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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = 0.01, size = 16, normalized size = 1.00 \[ -\frac {2 \sqrt {a+\frac {b}{x}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^2),x]

[Out]

(-2*Sqrt[a + b/x])/b

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 16, normalized size = 1.00 \[ -\frac {2 \, \sqrt {\frac {a x + b}{x}}}{b} \]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 16, normalized size = 1.00 \[ -\frac {2 \, \sqrt {\frac {a x + b}{x}}}{b} \]

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*x + b)/x)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 1.56 \[ -\frac {2 \left (a x +b \right )}{\sqrt {\frac {a x +b}{x}}\, b x} \]

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 = 1.14, size = 14, normalized size = 0.88 \[ -\frac {2 \, \sqrt {a + \frac {b}{x}}}{b} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.32, size = 14, normalized size = 0.88 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/x)^(1/2)),x)

[Out]

-(2*(a + b/x)^(1/2))/b

________________________________________________________________________________________

sympy [A]  time = 0.96, size = 22, normalized size = 1.38 \[ \begin {cases} - \frac {2 \sqrt {a + \frac {b}{x}}}{b} & \text {for}\: b \neq 0 \\- \frac {1}{\sqrt {a} x} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________