3.1577 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{a (a x+b)^4}{20 b^2 x^4}-\frac{(a x+b)^4}{5 b x^5} \]

[Out]

-(b + a*x)^4/(5*b*x^5) + (a*(b + a*x)^4)/(20*b^2*x^4)

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Rubi [A]  time = 0.0366419, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a x+b)^4}{20 b^2 x^4}-\frac{(a x+b)^4}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/x^3,x]

[Out]

-(b + a*x)^4/(5*b*x^5) + (a*(b + a*x)^4)/(20*b^2*x^4)

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Rubi in Sympy [A]  time = 7.83464, size = 37, normalized size = 1.03 \[ - \frac{a^{3}}{2 x^{2}} - \frac{a^{2} b}{x^{3}} - \frac{3 a b^{2}}{4 x^{4}} - \frac{b^{3}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00989323, size = 41, normalized size = 1.14 \[ -\frac{a^3}{2 x^2}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{4 x^4}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3/x^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 1. \[ -{\frac{3\,a{b}^{2}}{4\,{x}^{4}}}-{\frac{{a}^{2}b}{{x}^{3}}}-{\frac{{a}^{3}}{2\,{x}^{2}}}-{\frac{{b}^{3}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44363, size = 47, normalized size = 1.31 \[ -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5

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Fricas [A]  time = 0.211327, size = 47, normalized size = 1.31 \[ -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5

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Sympy [A]  time = 1.46644, size = 37, normalized size = 1.03 \[ - \frac{10 a^{3} x^{3} + 20 a^{2} b x^{2} + 15 a b^{2} x + 4 b^{3}}{20 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3/x**3,x)

[Out]

-(10*a**3*x**3 + 20*a**2*b*x**2 + 15*a*b**2*x + 4*b**3)/(20*x**5)

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GIAC/XCAS [A]  time = 0.222589, size = 47, normalized size = 1.31 \[ -\frac{10 \, a^{3} x^{3} + 20 \, a^{2} b x^{2} + 15 \, a b^{2} x + 4 \, b^{3}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(10*a^3*x^3 + 20*a^2*b*x^2 + 15*a*b^2*x + 4*b^3)/x^5