3.1823 \(\int \frac{\left (a+\frac{b}{x^2}\right )^2}{x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]

[Out]

-b^2/(5*x^5) - (2*a*b)/(3*x^3) - a^2/x

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Rubi [A]  time = 0.0417866, antiderivative size = 28, 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 \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2/x^2,x]

[Out]

-b^2/(5*x^5) - (2*a*b)/(3*x^3) - a^2/x

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Rubi in Sympy [A]  time = 6.96934, size = 24, normalized size = 0.86 \[ - \frac{a^{2}}{x} - \frac{2 a b}{3 x^{3}} - \frac{b^{2}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/x - 2*a*b/(3*x**3) - b**2/(5*x**5)

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Mathematica [A]  time = 0.00157272, size = 28, normalized size = 1. \[ -\frac{a^2}{x}-\frac{2 a b}{3 x^3}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2/x^2,x]

[Out]

-b^2/(5*x^5) - (2*a*b)/(3*x^3) - a^2/x

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Maple [A]  time = 0.007, size = 25, normalized size = 0.9 \[ -{\frac{{b}^{2}}{5\,{x}^{5}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2/x^2,x)

[Out]

-1/5*b^2/x^5-2/3*a*b/x^3-a^2/x

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Maxima [A]  time = 1.44009, size = 35, normalized size = 1.25 \[ -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(15*a^2*x^4 + 10*a*b*x^2 + 3*b^2)/x^5

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Fricas [A]  time = 0.21654, size = 35, normalized size = 1.25 \[ -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(15*a^2*x^4 + 10*a*b*x^2 + 3*b^2)/x^5

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Sympy [A]  time = 1.32736, size = 27, normalized size = 0.96 \[ - \frac{15 a^{2} x^{4} + 10 a b x^{2} + 3 b^{2}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2/x**2,x)

[Out]

-(15*a**2*x**4 + 10*a*b*x**2 + 3*b**2)/(15*x**5)

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GIAC/XCAS [A]  time = 0.229926, size = 35, normalized size = 1.25 \[ -\frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 3 \, b^{2}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(15*a^2*x^4 + 10*a*b*x^2 + 3*b^2)/x^5