3.93 \(\int \frac{(a+b x)^5}{x^{10}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{a^5}{9 x^9}-\frac{5 a^4 b}{8 x^8}-\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(8*x^8) - (10*a^3*b^2)/(7*x^7) - (5*a^2*b^3)/(3*x^6) -
(a*b^4)/x^5 - b^5/(4*x^4)

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Rubi [A]  time = 0.0519505, antiderivative size = 67, 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 \[ -\frac{a^5}{9 x^9}-\frac{5 a^4 b}{8 x^8}-\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(8*x^8) - (10*a^3*b^2)/(7*x^7) - (5*a^2*b^3)/(3*x^6) -
(a*b^4)/x^5 - b^5/(4*x^4)

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Rubi in Sympy [A]  time = 10.5387, size = 65, normalized size = 0.97 \[ - \frac{a^{5}}{9 x^{9}} - \frac{5 a^{4} b}{8 x^{8}} - \frac{10 a^{3} b^{2}}{7 x^{7}} - \frac{5 a^{2} b^{3}}{3 x^{6}} - \frac{a b^{4}}{x^{5}} - \frac{b^{5}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**10,x)

[Out]

-a**5/(9*x**9) - 5*a**4*b/(8*x**8) - 10*a**3*b**2/(7*x**7) - 5*a**2*b**3/(3*x**6
) - a*b**4/x**5 - b**5/(4*x**4)

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Mathematica [A]  time = 0.00928879, size = 67, normalized size = 1. \[ -\frac{a^5}{9 x^9}-\frac{5 a^4 b}{8 x^8}-\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(8*x^8) - (10*a^3*b^2)/(7*x^7) - (5*a^2*b^3)/(3*x^6) -
(a*b^4)/x^5 - b^5/(4*x^4)

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Maple [A]  time = 0.008, size = 58, normalized size = 0.9 \[ -{\frac{{a}^{5}}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}b}{8\,{x}^{8}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,{x}^{7}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,{x}^{6}}}-{\frac{a{b}^{4}}{{x}^{5}}}-{\frac{{b}^{5}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^10,x)

[Out]

-1/9*a^5/x^9-5/8*a^4*b/x^8-10/7*a^3*b^2/x^7-5/3*a^2*b^3/x^6-a*b^4/x^5-1/4*b^5/x^
4

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Maxima [A]  time = 1.33215, size = 77, normalized size = 1.15 \[ -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^
4*b*x + 56*a^5)/x^9

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Fricas [A]  time = 0.191349, size = 77, normalized size = 1.15 \[ -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^
4*b*x + 56*a^5)/x^9

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Sympy [A]  time = 2.20348, size = 61, normalized size = 0.91 \[ - \frac{56 a^{5} + 315 a^{4} b x + 720 a^{3} b^{2} x^{2} + 840 a^{2} b^{3} x^{3} + 504 a b^{4} x^{4} + 126 b^{5} x^{5}}{504 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**10,x)

[Out]

-(56*a**5 + 315*a**4*b*x + 720*a**3*b**2*x**2 + 840*a**2*b**3*x**3 + 504*a*b**4*
x**4 + 126*b**5*x**5)/(504*x**9)

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GIAC/XCAS [A]  time = 0.213393, size = 77, normalized size = 1.15 \[ -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^
4*b*x + 56*a^5)/x^9