3.1832 \(\int \left (a+\frac{b}{x^2}\right )^3 x^2 \, dx\)

Optimal. Leaf size=37 \[ \frac{a^3 x^3}{3}+3 a^2 b x-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.05034, antiderivative size = 37, 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 \[ \frac{a^3 x^3}{3}+3 a^2 b x-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.18191, size = 32, normalized size = 0.86 \[ \frac{a^{3} x^{3}}{3} + 3 a^{2} b x - \frac{3 a b^{2}}{x} - \frac{b^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00767255, size = 37, normalized size = 1. \[ \frac{a^3 x^3}{3}+3 a^2 b x-\frac{3 a b^2}{x}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3*x^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 34, normalized size = 0.9 \[ -{\frac{{b}^{3}}{3\,{x}^{3}}}-3\,{\frac{a{b}^{2}}{x}}+3\,{a}^{2}bx+{\frac{{a}^{3}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3*x^2,x)

[Out]

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

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Maxima [A]  time = 1.43807, size = 46, normalized size = 1.24 \[ \frac{1}{3} \, a^{3} x^{3} + 3 \, a^{2} b x - \frac{9 \, a b^{2} x^{2} + b^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 3*a^2*b*x - 1/3*(9*a*b^2*x^2 + b^3)/x^3

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Fricas [A]  time = 0.210652, size = 49, normalized size = 1.32 \[ \frac{a^{3} x^{6} + 9 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - b^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(a^3*x^6 + 9*a^2*b*x^4 - 9*a*b^2*x^2 - b^3)/x^3

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Sympy [A]  time = 1.24373, size = 34, normalized size = 0.92 \[ \frac{a^{3} x^{3}}{3} + 3 a^{2} b x - \frac{9 a b^{2} x^{2} + b^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3*x**2,x)

[Out]

a**3*x**3/3 + 3*a**2*b*x - (9*a*b**2*x**2 + b**3)/(3*x**3)

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GIAC/XCAS [A]  time = 0.232118, size = 46, normalized size = 1.24 \[ \frac{1}{3} \, a^{3} x^{3} + 3 \, a^{2} b x - \frac{9 \, a b^{2} x^{2} + b^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 3*a^2*b*x - 1/3*(9*a*b^2*x^2 + b^3)/x^3