3.1568 \(\int \left (a+\frac{b}{x}\right )^3 x^6 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^3 x^7}{7}+\frac{1}{2} a^2 b x^6+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

[Out]

(b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2 + (a^3*x^7)/7

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Rubi [A]  time = 0.0636702, 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 \[ \frac{a^3 x^7}{7}+\frac{1}{2} a^2 b x^6+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^6,x]

[Out]

(b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2 + (a^3*x^7)/7

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Rubi in Sympy [A]  time = 8.89244, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{7}}{7} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**7/7 + a**2*b*x**6/2 + 3*a*b**2*x**5/5 + b**3*x**4/4

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Mathematica [A]  time = 0.00308048, size = 43, normalized size = 1. \[ \frac{a^3 x^7}{7}+\frac{1}{2} a^2 b x^6+\frac{3}{5} a b^2 x^5+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x^6,x]

[Out]

(b^3*x^4)/4 + (3*a*b^2*x^5)/5 + (a^2*b*x^6)/2 + (a^3*x^7)/7

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{3\,a{b}^{2}{x}^{5}}{5}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{{a}^{3}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^6,x)

[Out]

1/4*b^3*x^4+3/5*a*b^2*x^5+1/2*a^2*b*x^6+1/7*a^3*x^7

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Maxima [A]  time = 1.43573, size = 47, normalized size = 1.09 \[ \frac{1}{7} \, a^{3} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4

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Fricas [A]  time = 0.211766, size = 47, normalized size = 1.09 \[ \frac{1}{7} \, a^{3} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4

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Sympy [A]  time = 0.099061, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{7}}{7} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x**6,x)

[Out]

a**3*x**7/7 + a**2*b*x**6/2 + 3*a*b**2*x**5/5 + b**3*x**4/4

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GIAC/XCAS [A]  time = 0.220484, size = 47, normalized size = 1.09 \[ \frac{1}{7} \, a^{3} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*a^3*x^7 + 1/2*a^2*b*x^6 + 3/5*a*b^2*x^5 + 1/4*b^3*x^4