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*(a + b/x)^(3/2))/(3*b)

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Rubi [A]  time = 0.0054849, antiderivative size = 18, 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} \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]

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*}

Mathematica [A]  time = 0.0063915, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]

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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Maple [A]  time = 0.003, size = 25, normalized size = 1.4 \begin{align*} -{\frac{2\,ax+2\,b}{3\,bx}\sqrt{{\frac{ax+b}{x}}}} \end{align*}

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 = 1.00931, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}

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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Fricas [A]  time = 1.67137, size = 54, normalized size = 3. \begin{align*} -\frac{2 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b x} \end{align*}

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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Sympy [B]  time = 1.10398, size = 41, normalized size = 2.28 \begin{align*} - \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} \end{align*}

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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Giac [B]  time = 1.12869, size = 112, normalized size = 6.22 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a \mathrm{sgn}\left (x\right ) + 3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b \mathrm{sgn}\left (x\right ) + b^{2} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3}} \end{align*}

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