3.1601 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^5} \, dx\)

Optimal. Leaf size=76 \[ \frac{a^3 (a x+b)^9}{1980 b^4 x^9}-\frac{a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac{a (a x+b)^9}{44 b^2 x^{11}}-\frac{(a x+b)^9}{12 b x^{12}} \]

[Out]

-(b + a*x)^9/(12*b*x^12) + (a*(b + a*x)^9)/(44*b^2*x^11) - (a^2*(b + a*x)^9)/(22
0*b^3*x^10) + (a^3*(b + a*x)^9)/(1980*b^4*x^9)

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Rubi [A]  time = 0.0760113, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^3 (a x+b)^9}{1980 b^4 x^9}-\frac{a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac{a (a x+b)^9}{44 b^2 x^{11}}-\frac{(a x+b)^9}{12 b x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^5,x]

[Out]

-(b + a*x)^9/(12*b*x^12) + (a*(b + a*x)^9)/(44*b^2*x^11) - (a^2*(b + a*x)^9)/(22
0*b^3*x^10) + (a^3*(b + a*x)^9)/(1980*b^4*x^9)

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Rubi in Sympy [A]  time = 11.1138, size = 66, normalized size = 0.87 \[ \frac{a^{3} \left (a x + b\right )^{9}}{1980 b^{4} x^{9}} - \frac{a^{2} \left (a x + b\right )^{9}}{220 b^{3} x^{10}} + \frac{a \left (a x + b\right )^{9}}{44 b^{2} x^{11}} - \frac{\left (a x + b\right )^{9}}{12 b x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8/x**5,x)

[Out]

a**3*(a*x + b)**9/(1980*b**4*x**9) - a**2*(a*x + b)**9/(220*b**3*x**10) + a*(a*x
+ b)**9/(44*b**2*x**11) - (a*x + b)**9/(12*b*x**12)

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Mathematica [A]  time = 0.00734809, size = 106, normalized size = 1.39 \[ -\frac{a^8}{4 x^4}-\frac{8 a^7 b}{5 x^5}-\frac{14 a^6 b^2}{3 x^6}-\frac{8 a^5 b^3}{x^7}-\frac{35 a^4 b^4}{4 x^8}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{5 x^{10}}-\frac{8 a b^7}{11 x^{11}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^5,x]

[Out]

-b^8/(12*x^12) - (8*a*b^7)/(11*x^11) - (14*a^2*b^6)/(5*x^10) - (56*a^3*b^5)/(9*x
^9) - (35*a^4*b^4)/(4*x^8) - (8*a^5*b^3)/x^7 - (14*a^6*b^2)/(3*x^6) - (8*a^7*b)/
(5*x^5) - a^8/(4*x^4)

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Maple [A]  time = 0.01, size = 91, normalized size = 1.2 \[ -{\frac{{b}^{8}}{12\,{x}^{12}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-{\frac{{a}^{8}}{4\,{x}^{4}}}-{\frac{14\,{a}^{2}{b}^{6}}{5\,{x}^{10}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,{x}^{8}}}-{\frac{8\,a{b}^{7}}{11\,{x}^{11}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}}-{\frac{8\,{a}^{7}b}{5\,{x}^{5}}}-8\,{\frac{{a}^{5}{b}^{3}}{{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^5,x)

[Out]

-1/12*b^8/x^12-14/3*a^6*b^2/x^6-1/4*a^8/x^4-14/5*a^2*b^6/x^10-35/4*a^4*b^4/x^8-8
/11*a*b^7/x^11-56/9*a^3*b^5/x^9-8/5*a^7*b/x^5-8*a^5*b^3/x^7

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Maxima [A]  time = 1.44047, size = 122, normalized size = 1.61 \[ -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^5,x, algorithm="maxima")

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 1
7325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8
)/x^12

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Fricas [A]  time = 0.216908, size = 122, normalized size = 1.61 \[ -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^5,x, algorithm="fricas")

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 1
7325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8
)/x^12

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Sympy [A]  time = 3.44365, size = 97, normalized size = 1.28 \[ - \frac{495 a^{8} x^{8} + 3168 a^{7} b x^{7} + 9240 a^{6} b^{2} x^{6} + 15840 a^{5} b^{3} x^{5} + 17325 a^{4} b^{4} x^{4} + 12320 a^{3} b^{5} x^{3} + 5544 a^{2} b^{6} x^{2} + 1440 a b^{7} x + 165 b^{8}}{1980 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**5,x)

[Out]

-(495*a**8*x**8 + 3168*a**7*b*x**7 + 9240*a**6*b**2*x**6 + 15840*a**5*b**3*x**5
+ 17325*a**4*b**4*x**4 + 12320*a**3*b**5*x**3 + 5544*a**2*b**6*x**2 + 1440*a*b**
7*x + 165*b**8)/(1980*x**12)

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GIAC/XCAS [A]  time = 0.221204, size = 122, normalized size = 1.61 \[ -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^5,x, algorithm="giac")

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 1
7325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8
)/x^12