3.1821 \(\int \left (a+\frac{b}{x^2}\right )^2 \, dx\)

Optimal. Leaf size=23 \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

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Rubi [A]  time = 0.0334907, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2,x]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a b}{x} - \frac{b^{2}}{3 x^{3}} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.00878449, size = 23, normalized size = 1. \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2,x]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

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Maple [A]  time = 0.008, size = 22, normalized size = 1. \[ -{\frac{{b}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}}+x{a}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.43518, size = 28, normalized size = 1.22 \[ a^{2} x - \frac{2 \, a b}{x} - \frac{b^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x - 2*a*b/x - 1/3*b^2/x^3

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Fricas [A]  time = 0.212713, size = 35, normalized size = 1.52 \[ \frac{3 \, a^{2} x^{4} - 6 \, a b x^{2} - b^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*a^2*x^4 - 6*a*b*x^2 - b^2)/x^3

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Sympy [A]  time = 1.17635, size = 20, normalized size = 0.87 \[ a^{2} x - \frac{6 a b x^{2} + b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2,x)

[Out]

a**2*x - (6*a*b*x**2 + b**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.227513, size = 30, normalized size = 1.3 \[ a^{2} x - \frac{6 \, a b x^{2} + b^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x - 1/3*(6*a*b*x^2 + b^2)/x^3