3.1696 \(\int \frac {\sqrt {a+\frac {b}{x}}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, 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} \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/x^2,x]

[Out]

(-2*(a + b/x)^(3/2))/(3*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 {\sqrt {a+\frac {b}{x}}}{x^2} \, dx &=-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/x^2,x]

[Out]

(-2*(a + b/x)^(3/2))/(3*b)

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fricas [A]  time = 1.19, size = 24, normalized size = 1.33 \[ -\frac {2 \, {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{3 \, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(a*x + b)*sqrt((a*x + b)/x)/(b*x)

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giac [B]  time = 0.22, size = 83, normalized size = 4.61 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a \mathrm {sgn}\relax (x) + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b \mathrm {sgn}\relax (x) + b^{2} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*sgn(x) + 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b*sgn(x) + b^2*s
gn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^3

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maple [A]  time = 0.00, size = 25, normalized size = 1.39 \[ -\frac {2 \left (a x +b \right ) \sqrt {\frac {a x +b}{x}}}{3 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(a + b/x)^(3/2)/b

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mupad [B]  time = 1.37, size = 22, normalized size = 1.22 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (b+a\,x\right )}{3\,b\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(1/2)/x^2,x)

[Out]

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

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sympy [B]  time = 0.94, size = 41, normalized size = 2.28 \[ - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}}{3 b} - \frac {2 \sqrt {a} \sqrt {1 + \frac {b}{a x}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x))/(3*b) - 2*sqrt(a)*sqrt(1 + b/(a*x))/(3*x)

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