3.1830 \(\int \left (a+\frac{b}{x^2}\right )^3 x^4 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^3 x^5}{5}+a^2 b x^3+3 a b^2 x-\frac{b^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0494339, antiderivative size = 34, 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 \[ \frac{a^3 x^5}{5}+a^2 b x^3+3 a b^2 x-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3*x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.07824, size = 29, normalized size = 0.85 \[ \frac{a^{3} x^{5}}{5} + a^{2} b x^{3} + 3 a b^{2} x - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**3*x**4,x)

[Out]

a**3*x**5/5 + a**2*b*x**3 + 3*a*b**2*x - b**3/x

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Mathematica [A]  time = 0.00721594, size = 34, normalized size = 1. \[ \frac{a^3 x^5}{5}+a^2 b x^3+3 a b^2 x-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3*x^4,x]

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 1. \[ -{\frac{{b}^{3}}{x}}+3\,a{b}^{2}x+{a}^{2}b{x}^{3}+{\frac{{a}^{3}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3*x^4,x)

[Out]

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

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Maxima [A]  time = 1.44086, size = 43, normalized size = 1.26 \[ \frac{1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215482, size = 49, normalized size = 1.44 \[ \frac{a^{3} x^{6} + 5 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} - 5 \, b^{3}}{5 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3*x^4,x, algorithm="fricas")

[Out]

1/5*(a^3*x^6 + 5*a^2*b*x^4 + 15*a*b^2*x^2 - 5*b^3)/x

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Sympy [A]  time = 1.09121, size = 29, normalized size = 0.85 \[ \frac{a^{3} x^{5}}{5} + a^{2} b x^{3} + 3 a b^{2} x - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3*x**4,x)

[Out]

a**3*x**5/5 + a**2*b*x**3 + 3*a*b**2*x - b**3/x

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GIAC/XCAS [A]  time = 0.22815, size = 43, normalized size = 1.26 \[ \frac{1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3*x^4,x, algorithm="giac")

[Out]

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