3.1602 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^6} \, dx\)

Optimal. Leaf size=96 \[ -\frac{a^4 (a x+b)^9}{6435 b^5 x^9}+\frac{a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac{a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac{a (a x+b)^9}{39 b^2 x^{12}}-\frac{(a x+b)^9}{13 b x^{13}} \]

[Out]

-(b + a*x)^9/(13*b*x^13) + (a*(b + a*x)^9)/(39*b^2*x^12) - (a^2*(b + a*x)^9)/(14
3*b^3*x^11) + (a^3*(b + a*x)^9)/(715*b^4*x^10) - (a^4*(b + a*x)^9)/(6435*b^5*x^9
)

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Rubi [A]  time = 0.101826, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{a^4 (a x+b)^9}{6435 b^5 x^9}+\frac{a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac{a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac{a (a x+b)^9}{39 b^2 x^{12}}-\frac{(a x+b)^9}{13 b x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^6,x]

[Out]

-(b + a*x)^9/(13*b*x^13) + (a*(b + a*x)^9)/(39*b^2*x^12) - (a^2*(b + a*x)^9)/(14
3*b^3*x^11) + (a^3*(b + a*x)^9)/(715*b^4*x^10) - (a^4*(b + a*x)^9)/(6435*b^5*x^9
)

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Rubi in Sympy [A]  time = 20.4684, size = 105, normalized size = 1.09 \[ - \frac{a^{8}}{5 x^{5}} - \frac{4 a^{7} b}{3 x^{6}} - \frac{4 a^{6} b^{2}}{x^{7}} - \frac{7 a^{5} b^{3}}{x^{8}} - \frac{70 a^{4} b^{4}}{9 x^{9}} - \frac{28 a^{3} b^{5}}{5 x^{10}} - \frac{28 a^{2} b^{6}}{11 x^{11}} - \frac{2 a b^{7}}{3 x^{12}} - \frac{b^{8}}{13 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8/x**6,x)

[Out]

-a**8/(5*x**5) - 4*a**7*b/(3*x**6) - 4*a**6*b**2/x**7 - 7*a**5*b**3/x**8 - 70*a*
*4*b**4/(9*x**9) - 28*a**3*b**5/(5*x**10) - 28*a**2*b**6/(11*x**11) - 2*a*b**7/(
3*x**12) - b**8/(13*x**13)

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Mathematica [A]  time = 0.0155441, size = 104, normalized size = 1.08 \[ -\frac{a^8}{5 x^5}-\frac{4 a^7 b}{3 x^6}-\frac{4 a^6 b^2}{x^7}-\frac{7 a^5 b^3}{x^8}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{5 x^{10}}-\frac{28 a^2 b^6}{11 x^{11}}-\frac{2 a b^7}{3 x^{12}}-\frac{b^8}{13 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^6,x]

[Out]

-b^8/(13*x^13) - (2*a*b^7)/(3*x^12) - (28*a^2*b^6)/(11*x^11) - (28*a^3*b^5)/(5*x
^10) - (70*a^4*b^4)/(9*x^9) - (7*a^5*b^3)/x^8 - (4*a^6*b^2)/x^7 - (4*a^7*b)/(3*x
^6) - a^8/(5*x^5)

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Maple [A]  time = 0.009, size = 91, normalized size = 1. \[ -{\frac{2\,a{b}^{7}}{3\,{x}^{12}}}-{\frac{{b}^{8}}{13\,{x}^{13}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,{x}^{10}}}-7\,{\frac{{a}^{5}{b}^{3}}{{x}^{8}}}-{\frac{28\,{a}^{2}{b}^{6}}{11\,{x}^{11}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,{x}^{9}}}-{\frac{{a}^{8}}{5\,{x}^{5}}}-4\,{\frac{{a}^{6}{b}^{2}}{{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^6,x)

[Out]

-2/3*a*b^7/x^12-1/13*b^8/x^13-4/3*a^7*b/x^6-28/5*a^3*b^5/x^10-7*a^5*b^3/x^8-28/1
1*a^2*b^6/x^11-70/9*a^4*b^4/x^9-1/5*a^8/x^5-4*a^6*b^2/x^7

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Maxima [A]  time = 1.45311, size = 122, normalized size = 1.27 \[ -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^6,x, algorithm="maxima")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 +
50050*a^4*b^4*x^4 + 36036*a^3*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*
b^8)/x^13

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Fricas [A]  time = 0.211146, size = 122, normalized size = 1.27 \[ -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^6,x, algorithm="fricas")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 +
50050*a^4*b^4*x^4 + 36036*a^3*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*
b^8)/x^13

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Sympy [A]  time = 3.63455, size = 97, normalized size = 1.01 \[ - \frac{1287 a^{8} x^{8} + 8580 a^{7} b x^{7} + 25740 a^{6} b^{2} x^{6} + 45045 a^{5} b^{3} x^{5} + 50050 a^{4} b^{4} x^{4} + 36036 a^{3} b^{5} x^{3} + 16380 a^{2} b^{6} x^{2} + 4290 a b^{7} x + 495 b^{8}}{6435 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**6,x)

[Out]

-(1287*a**8*x**8 + 8580*a**7*b*x**7 + 25740*a**6*b**2*x**6 + 45045*a**5*b**3*x**
5 + 50050*a**4*b**4*x**4 + 36036*a**3*b**5*x**3 + 16380*a**2*b**6*x**2 + 4290*a*
b**7*x + 495*b**8)/(6435*x**13)

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GIAC/XCAS [A]  time = 0.221118, size = 122, normalized size = 1.27 \[ -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^6,x, algorithm="giac")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 +
50050*a^4*b^4*x^4 + 36036*a^3*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*
b^8)/x^13