3.1811 \(\int \frac{a+\frac{b}{x^2}}{x^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

[Out]

-b/(4*x^4) - a/(2*x^2)

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Rubi [A]  time = 0.0150459, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/x^3,x]

[Out]

-b/(4*x^4) - a/(2*x^2)

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Rubi in Sympy [A]  time = 2.87073, size = 14, normalized size = 0.82 \[ - \frac{a}{2 x^{2}} - \frac{b}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*x**2) - b/(4*x**4)

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Mathematica [A]  time = 0.00383628, size = 17, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/x^3,x]

[Out]

-b/(4*x^4) - a/(2*x^2)

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Maple [A]  time = 0.006, size = 14, normalized size = 0.8 \[ -{\frac{b}{4\,{x}^{4}}}-{\frac{a}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44467, size = 19, normalized size = 1.12 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{2}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a + b/x^2)^2/b

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Fricas [A]  time = 0.218902, size = 18, normalized size = 1.06 \[ -\frac{2 \, a x^{2} + b}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*a*x^2 + b)/x^4

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Sympy [A]  time = 1.12329, size = 14, normalized size = 0.82 \[ - \frac{2 a x^{2} + b}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/x**3,x)

[Out]

-(2*a*x**2 + b)/(4*x**4)

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GIAC/XCAS [A]  time = 0.22647, size = 18, normalized size = 1.06 \[ -\frac{2 \, a x^{2} + b}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*a*x^2 + b)/x^4