3.1816 \(\int \left (a+\frac{b}{x^2}\right )^2 x^5 \, dx\)

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

[Out]

(b + a*x^2)^3/(6*a)

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Rubi [A]  time = 0.0212066, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (a x^2+b\right )^3}{6 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*x^5,x]

[Out]

(b + a*x^2)^3/(6*a)

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Rubi in Sympy [A]  time = 3.4856, size = 10, normalized size = 0.62 \[ \frac{\left (a x^{2} + b\right )^{3}}{6 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**2*x**5,x)

[Out]

(a*x**2 + b)**3/(6*a)

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Mathematica [A]  time = 0.00385963, size = 16, normalized size = 1. \[ \frac{\left (a x^2+b\right )^3}{6 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*x^5,x]

[Out]

(b + a*x^2)^3/(6*a)

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Maple [A]  time = 0.001, size = 25, normalized size = 1.6 \[{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*x^5,x)

[Out]

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

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Maxima [A]  time = 1.44141, size = 32, normalized size = 2. \[ \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.218408, size = 32, normalized size = 2. \[ \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2*x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.088102, size = 24, normalized size = 1.5 \[ \frac{a^{2} x^{6}}{6} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*x**5,x)

[Out]

a**2*x**6/6 + a*b*x**4/2 + b**2*x**2/2

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GIAC/XCAS [A]  time = 0.226538, size = 32, normalized size = 2. \[ \frac{1}{6} \, a^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2*x^5,x, algorithm="giac")

[Out]

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