3.76 \(\int \frac{(a+b x)^3}{x^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03139, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^3}{7 x^7}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.35112, size = 39, normalized size = 0.91 \[ - \frac{a^{3}}{7 x^{7}} - \frac{a^{2} b}{2 x^{6}} - \frac{3 a b^{2}}{5 x^{5}} - \frac{b^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.00457256, size = 43, normalized size = 1. \[ -\frac{a^3}{7 x^7}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/x^8,x]

[Out]

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

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \[ -{\frac{{a}^{3}}{7\,{x}^{7}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{3\,a{b}^{2}}{5\,{x}^{5}}}-{\frac{{b}^{3}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/x^8,x)

[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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Maxima [A]  time = 1.33498, size = 47, normalized size = 1.09 \[ -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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Fricas [A]  time = 0.190005, size = 47, normalized size = 1.09 \[ -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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Sympy [A]  time = 1.57391, size = 37, normalized size = 0.86 \[ - \frac{20 a^{3} + 70 a^{2} b x + 84 a b^{2} x^{2} + 35 b^{3} x^{3}}{140 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/x**8,x)

[Out]

-(20*a**3 + 70*a**2*b*x + 84*a*b**2*x**2 + 35*b**3*x**3)/(140*x**7)

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GIAC/XCAS [A]  time = 0.213229, size = 47, normalized size = 1.09 \[ -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/x^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7