3.1564 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x}\right )^3}{3 b} \]

[Out]

-(a + b/x)^3/(3*b)

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Rubi [A]  time = 0.0165953, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\left (a+\frac{b}{x}\right )^3}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x^2,x]

[Out]

-(a + b/x)^3/(3*b)

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Rubi in Sympy [A]  time = 2.21854, size = 10, normalized size = 0.62 \[ - \frac{\left (a + \frac{b}{x}\right )^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2/x**2,x)

[Out]

-(a + b/x)**3/(3*b)

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Mathematica [A]  time = 0.0106398, size = 26, normalized size = 1.62 \[ -\frac{a^2}{x}-\frac{a b}{x^2}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2/x^2,x]

[Out]

-b^2/(3*x^3) - (a*b)/x^2 - a^2/x

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Maple [A]  time = 0.008, size = 25, normalized size = 1.6 \[ -{\frac{{b}^{2}}{3\,{x}^{3}}}-{\frac{ab}{{x}^{2}}}-{\frac{{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x^2,x)

[Out]

-1/3*b^2/x^3-a*b/x^2-a^2/x

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Maxima [A]  time = 1.44281, size = 19, normalized size = 1.19 \[ -\frac{{\left (a + \frac{b}{x}\right )}^{3}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(a + b/x)^3/b

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Fricas [A]  time = 0.216595, size = 30, normalized size = 1.88 \[ -\frac{3 \, a^{2} x^{2} + 3 \, a b x + b^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*a^2*x^2 + 3*a*b*x + b^2)/x^3

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Sympy [A]  time = 1.20217, size = 24, normalized size = 1.5 \[ - \frac{3 a^{2} x^{2} + 3 a b x + b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226408, size = 19, normalized size = 1.19 \[ -\frac{{\left (a + \frac{b}{x}\right )}^{3}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(a + b/x)^3/b