3.1578 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{x^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

[Out]

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

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Rubi [A]  time = 0.0454398, 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 \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.95168, size = 41, normalized size = 0.95 \[ - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{4 x^{4}} - \frac{3 a b^{2}}{5 x^{5}} - \frac{b^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(3*x**3) - 3*a**2*b/(4*x**4) - 3*a*b**2/(5*x**5) - b**3/(6*x**6)

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Mathematica [A]  time = 0.00569218, size = 43, normalized size = 1. \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \[ -{\frac{{b}^{3}}{6\,{x}^{6}}}-{\frac{3\,a{b}^{2}}{5\,{x}^{5}}}-{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{{a}^{3}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^4,x)

[Out]

-1/6*b^3/x^6-3/5*a*b^2/x^5-3/4*a^2*b/x^4-1/3*a^3/x^3

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Maxima [A]  time = 1.43363, size = 47, normalized size = 1.09 \[ -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6

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Fricas [A]  time = 0.215967, size = 47, normalized size = 1.09 \[ -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6

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Sympy [A]  time = 1.53503, size = 37, normalized size = 0.86 \[ - \frac{20 a^{3} x^{3} + 45 a^{2} b x^{2} + 36 a b^{2} x + 10 b^{3}}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3/x**4,x)

[Out]

-(20*a**3*x**3 + 45*a**2*b*x**2 + 36*a*b**2*x + 10*b**3)/(60*x**6)

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GIAC/XCAS [A]  time = 0.21873, size = 47, normalized size = 1.09 \[ -\frac{20 \, a^{3} x^{3} + 45 \, a^{2} b x^{2} + 36 \, a b^{2} x + 10 \, b^{3}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*a^3*x^3 + 45*a^2*b*x^2 + 36*a*b^2*x + 10*b^3)/x^6