3.1824 \(\int \frac{\left (a+\frac{b}{x^2}\right )^2}{x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x^2}\right )^3}{6 b} \]

[Out]

-(a + b/x^2)^3/(6*b)

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Rubi [A]  time = 0.0188534, 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 \[ -\frac{\left (a+\frac{b}{x^2}\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2/x^3,x]

[Out]

-(a + b/x^2)^3/(6*b)

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Rubi in Sympy [A]  time = 2.13121, size = 12, normalized size = 0.75 \[ - \frac{\left (a + \frac{b}{x^{2}}\right )^{3}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b/x**2)**3/(6*b)

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Mathematica [A]  time = 0.00160343, size = 30, normalized size = 1.88 \[ -\frac{a^2}{2 x^2}-\frac{a b}{2 x^4}-\frac{b^2}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2/x^3,x]

[Out]

-b^2/(6*x^6) - (a*b)/(2*x^4) - a^2/(2*x^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 1.6 \[ -{\frac{{b}^{2}}{6\,{x}^{6}}}-{\frac{ab}{2\,{x}^{4}}}-{\frac{{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2/x^3,x)

[Out]

-1/6*b^2/x^6-1/2*a*b/x^4-1/2*a^2/x^2

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Maxima [A]  time = 1.4382, size = 19, normalized size = 1.19 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{3}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(a + b/x^2)^3/b

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Fricas [A]  time = 0.219369, size = 32, normalized size = 2. \[ -\frac{3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.33933, size = 26, normalized size = 1.62 \[ - \frac{3 a^{2} x^{4} + 3 a b x^{2} + b^{2}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226394, size = 32, normalized size = 2. \[ -\frac{3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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