3.92 \(\int \frac{(a+b x)^5}{x^9} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^6}{168 a^3 x^6}+\frac{b (a+b x)^6}{28 a^2 x^7}-\frac{(a+b x)^6}{8 a x^8} \]

[Out]

-(a + b*x)^6/(8*a*x^8) + (b*(a + b*x)^6)/(28*a^2*x^7) - (b^2*(a + b*x)^6)/(168*a
^3*x^6)

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Rubi [A]  time = 0.0383496, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^2 (a+b x)^6}{168 a^3 x^6}+\frac{b (a+b x)^6}{28 a^2 x^7}-\frac{(a+b x)^6}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^9,x]

[Out]

-(a + b*x)^6/(8*a*x^8) + (b*(a + b*x)^6)/(28*a^2*x^7) - (b^2*(a + b*x)^6)/(168*a
^3*x^6)

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Rubi in Sympy [A]  time = 10.4682, size = 66, normalized size = 1.18 \[ - \frac{a^{5}}{8 x^{8}} - \frac{5 a^{4} b}{7 x^{7}} - \frac{5 a^{3} b^{2}}{3 x^{6}} - \frac{2 a^{2} b^{3}}{x^{5}} - \frac{5 a b^{4}}{4 x^{4}} - \frac{b^{5}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**9,x)

[Out]

-a**5/(8*x**8) - 5*a**4*b/(7*x**7) - 5*a**3*b**2/(3*x**6) - 2*a**2*b**3/x**5 - 5
*a*b**4/(4*x**4) - b**5/(3*x**3)

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Mathematica [A]  time = 0.00540067, size = 67, normalized size = 1.2 \[ -\frac{a^5}{8 x^8}-\frac{5 a^4 b}{7 x^7}-\frac{5 a^3 b^2}{3 x^6}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{4 x^4}-\frac{b^5}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^9,x]

[Out]

-a^5/(8*x^8) - (5*a^4*b)/(7*x^7) - (5*a^3*b^2)/(3*x^6) - (2*a^2*b^3)/x^5 - (5*a*
b^4)/(4*x^4) - b^5/(3*x^3)

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Maple [A]  time = 0.008, size = 58, normalized size = 1. \[ -{\frac{{a}^{5}}{8\,{x}^{8}}}-2\,{\frac{{a}^{2}{b}^{3}}{{x}^{5}}}-{\frac{{b}^{5}}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}b}{7\,{x}^{7}}}-{\frac{5\,a{b}^{4}}{4\,{x}^{4}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^9,x)

[Out]

-1/8*a^5/x^8-2*a^2*b^3/x^5-1/3*b^5/x^3-5/7*a^4*b/x^7-5/4*a*b^4/x^4-5/3*a^3*b^2/x
^6

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Maxima [A]  time = 1.32657, size = 77, normalized size = 1.38 \[ -\frac{56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^9,x, algorithm="maxima")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4
*b*x + 21*a^5)/x^8

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Fricas [A]  time = 0.191469, size = 77, normalized size = 1.38 \[ -\frac{56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4
*b*x + 21*a^5)/x^8

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Sympy [A]  time = 2.12424, size = 61, normalized size = 1.09 \[ - \frac{21 a^{5} + 120 a^{4} b x + 280 a^{3} b^{2} x^{2} + 336 a^{2} b^{3} x^{3} + 210 a b^{4} x^{4} + 56 b^{5} x^{5}}{168 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**9,x)

[Out]

-(21*a**5 + 120*a**4*b*x + 280*a**3*b**2*x**2 + 336*a**2*b**3*x**3 + 210*a*b**4*
x**4 + 56*b**5*x**5)/(168*x**8)

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GIAC/XCAS [A]  time = 0.213959, size = 77, normalized size = 1.38 \[ -\frac{56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^9,x, algorithm="giac")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4
*b*x + 21*a^5)/x^8