3.1569 \(\int \left (a+\frac{b}{x}\right )^3 x^5 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0582798, 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 \[ \frac{a^3 x^6}{6}+\frac{3}{5} a^2 b x^5+\frac{3}{4} a b^2 x^4+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.44488, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3*x**5,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**5/5 + 3*a*b**2*x**4/4 + b**3*x**3/3

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Mathematica [A]  time = 0.00322767, size = 43, normalized size = 1. \[ \frac{a^3 x^6}{6}+\frac{3}{5} a^2 b x^5+\frac{3}{4} a b^2 x^4+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x^5,x]

[Out]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \[{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{{a}^{3}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^5,x)

[Out]

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

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Maxima [A]  time = 1.4344, size = 47, normalized size = 1.09 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216324, size = 47, normalized size = 1.09 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.097185, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x**5,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**5/5 + 3*a*b**2*x**4/4 + b**3*x**3/3

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GIAC/XCAS [A]  time = 0.226684, size = 47, normalized size = 1.09 \[ \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^5,x, algorithm="giac")

[Out]

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