3.1694 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^2} \, dx\)

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 = 0.0259538, 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 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^2,x]

[Out]

(-2*(a + b/x)^(3/2))/(3*b)

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Rubi in Sympy [A]  time = 2.22485, 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]  rubi_integrate((a+b/x)**(1/2)/x**2,x)

[Out]

-2*(a + b/x)**(3/2)/(3*b)

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Mathematica [A]  time = 0.0167095, 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 = 0.008, size = 25, normalized size = 1.4 \[ -{\frac{2\,ax+2\,b}{3\,bx}\sqrt{{\frac{ax+b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43968, size = 19, normalized size = 1.06 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^2,x, algorithm="maxima")

[Out]

-2/3*(a + b/x)^(3/2)/b

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Fricas [A]  time = 0.222922, size = 32, normalized size = 1.78 \[ -\frac{2 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.99631, 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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GIAC/XCAS [A]  time = 0.24907, size = 112, normalized size = 6.22 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a{\rm sign}\left (x\right ) + 3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b{\rm sign}\left (x\right ) + b^{2}{\rm sign}\left (x\right )\right )}}{3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^2,x, algorithm="giac")

[Out]

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