3.1819 \(\int \left (a+\frac{b}{x^2}\right )^2 x^2 \, dx\)

Optimal. Leaf size=24 \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

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Rubi [A]  time = 0.0404836, antiderivative size = 24, 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 \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*x^2,x]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

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Rubi in Sympy [A]  time = 6.64783, size = 19, normalized size = 0.79 \[ \frac{a^{2} x^{3}}{3} + 2 a b x - \frac{b^{2}}{x} \]

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**3/3 + 2*a*b*x - b**2/x

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Mathematica [A]  time = 0.00180374, size = 24, normalized size = 1. \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*x^2,x]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

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Maple [A]  time = 0.005, size = 23, normalized size = 1. \[ -{\frac{{b}^{2}}{x}}+2\,abx+{\frac{{x}^{3}{a}^{2}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*x^2,x)

[Out]

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

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Maxima [A]  time = 1.44128, size = 30, normalized size = 1.25 \[ \frac{1}{3} \, a^{2} x^{3} + 2 \, a b x - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + 2*a*b*x - b^2/x

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Fricas [A]  time = 0.220409, size = 34, normalized size = 1.42 \[ \frac{a^{2} x^{4} + 6 \, a b x^{2} - 3 \, b^{2}}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a^2*x^4 + 6*a*b*x^2 - 3*b^2)/x

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Sympy [A]  time = 1.09608, size = 19, normalized size = 0.79 \[ \frac{a^{2} x^{3}}{3} + 2 a b x - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*x**2,x)

[Out]

a**2*x**3/3 + 2*a*b*x - b**2/x

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GIAC/XCAS [A]  time = 0.226212, size = 30, normalized size = 1.25 \[ \frac{1}{3} \, a^{2} x^{3} + 2 \, a b x - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + 2*a*b*x - b^2/x