3.1565 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]

[Out]

-b^2/(4*x^4) - (2*a*b)/(3*x^3) - a^2/(2*x^2)

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Rubi [A]  time = 0.0366121, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x^3,x]

[Out]

-b^2/(4*x^4) - (2*a*b)/(3*x^3) - a^2/(2*x^2)

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Rubi in Sympy [A]  time = 6.16289, size = 27, normalized size = 0.9 \[ - \frac{a^{2}}{2 x^{2}} - \frac{2 a b}{3 x^{3}} - \frac{b^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2/x**3,x)

[Out]

-a**2/(2*x**2) - 2*a*b/(3*x**3) - b**2/(4*x**4)

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Mathematica [A]  time = 0.00567778, size = 30, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2/x^3,x]

[Out]

-b^2/(4*x^4) - (2*a*b)/(3*x^3) - a^2/(2*x^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.8 \[ -{\frac{{b}^{2}}{4\,{x}^{4}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x^3,x)

[Out]

-1/4*b^2/x^4-2/3*a*b/x^3-1/2*a^2/x^2

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Maxima [A]  time = 1.44055, size = 32, normalized size = 1.07 \[ -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^3,x, algorithm="maxima")

[Out]

-1/12*(6*a^2*x^2 + 8*a*b*x + 3*b^2)/x^4

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Fricas [A]  time = 0.219425, size = 32, normalized size = 1.07 \[ -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^3,x, algorithm="fricas")

[Out]

-1/12*(6*a^2*x^2 + 8*a*b*x + 3*b^2)/x^4

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Sympy [A]  time = 1.24637, size = 26, normalized size = 0.87 \[ - \frac{6 a^{2} x^{2} + 8 a b x + 3 b^{2}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2/x**3,x)

[Out]

-(6*a**2*x**2 + 8*a*b*x + 3*b**2)/(12*x**4)

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GIAC/XCAS [A]  time = 0.227715, size = 32, normalized size = 1.07 \[ -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^3,x, algorithm="giac")

[Out]

-1/12*(6*a^2*x^2 + 8*a*b*x + 3*b^2)/x^4