3.1580 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{x^6} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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Rubi [A]  time = 0.0467866, antiderivative size = 43, 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 \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/x^6,x]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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Rubi in Sympy [A]  time = 8.24555, size = 39, normalized size = 0.91 \[ - \frac{a^{3}}{5 x^{5}} - \frac{a^{2} b}{2 x^{6}} - \frac{3 a b^{2}}{7 x^{7}} - \frac{b^{3}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3/x**6,x)

[Out]

-a**3/(5*x**5) - a**2*b/(2*x**6) - 3*a*b**2/(7*x**7) - b**3/(8*x**8)

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Mathematica [A]  time = 0.0103863, size = 43, normalized size = 1. \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3/x^6,x]

[Out]

-b^3/(8*x^8) - (3*a*b^2)/(7*x^7) - (a^2*b)/(2*x^6) - a^3/(5*x^5)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[ -{\frac{{b}^{3}}{8\,{x}^{8}}}-{\frac{3\,a{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{{a}^{3}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^6,x)

[Out]

-1/8*b^3/x^8-3/7*a*b^2/x^7-1/2*a^2*b/x^6-1/5*a^3/x^5

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Maxima [A]  time = 1.45743, size = 47, normalized size = 1.09 \[ -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^6,x, algorithm="maxima")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8

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Fricas [A]  time = 0.210364, size = 47, normalized size = 1.09 \[ -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^6,x, algorithm="fricas")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8

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Sympy [A]  time = 1.62728, size = 37, normalized size = 0.86 \[ - \frac{56 a^{3} x^{3} + 140 a^{2} b x^{2} + 120 a b^{2} x + 35 b^{3}}{280 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3/x**6,x)

[Out]

-(56*a**3*x**3 + 140*a**2*b*x**2 + 120*a*b**2*x + 35*b**3)/(280*x**8)

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GIAC/XCAS [A]  time = 0.217919, size = 47, normalized size = 1.09 \[ -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^6,x, algorithm="giac")

[Out]

-1/280*(56*a^3*x^3 + 140*a^2*b*x^2 + 120*a*b^2*x + 35*b^3)/x^8