3.1603 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^7} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{7 x^7}-\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]

[Out]

-b^8/(14*x^14) - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5)/(11*x
^11) - (7*a^4*b^4)/x^10 - (56*a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)
/(7*x^7) - a^8/(6*x^6)

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Rubi [A]  time = 0.111275, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{7 x^7}-\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^7,x]

[Out]

-b^8/(14*x^14) - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5)/(11*x
^11) - (7*a^4*b^4)/x^10 - (56*a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)
/(7*x^7) - a^8/(6*x^6)

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Rubi in Sympy [A]  time = 20.4514, size = 107, normalized size = 1.01 \[ - \frac{a^{8}}{6 x^{6}} - \frac{8 a^{7} b}{7 x^{7}} - \frac{7 a^{6} b^{2}}{2 x^{8}} - \frac{56 a^{5} b^{3}}{9 x^{9}} - \frac{7 a^{4} b^{4}}{x^{10}} - \frac{56 a^{3} b^{5}}{11 x^{11}} - \frac{7 a^{2} b^{6}}{3 x^{12}} - \frac{8 a b^{7}}{13 x^{13}} - \frac{b^{8}}{14 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8/x**7,x)

[Out]

-a**8/(6*x**6) - 8*a**7*b/(7*x**7) - 7*a**6*b**2/(2*x**8) - 56*a**5*b**3/(9*x**9
) - 7*a**4*b**4/x**10 - 56*a**3*b**5/(11*x**11) - 7*a**2*b**6/(3*x**12) - 8*a*b*
*7/(13*x**13) - b**8/(14*x**14)

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Mathematica [A]  time = 0.00741721, size = 106, normalized size = 1. \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{7 x^7}-\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^7,x]

[Out]

-b^8/(14*x^14) - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5)/(11*x
^11) - (7*a^4*b^4)/x^10 - (56*a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)
/(7*x^7) - a^8/(6*x^6)

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Maple [A]  time = 0.008, size = 91, normalized size = 0.9 \[ -{\frac{{b}^{8}}{14\,{x}^{14}}}-{\frac{8\,a{b}^{7}}{13\,{x}^{13}}}-{\frac{7\,{a}^{2}{b}^{6}}{3\,{x}^{12}}}-{\frac{56\,{a}^{3}{b}^{5}}{11\,{x}^{11}}}-7\,{\frac{{a}^{4}{b}^{4}}{{x}^{10}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,{x}^{9}}}-{\frac{7\,{a}^{6}{b}^{2}}{2\,{x}^{8}}}-{\frac{8\,{a}^{7}b}{7\,{x}^{7}}}-{\frac{{a}^{8}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^7,x)

[Out]

-1/14*b^8/x^14-8/13*a*b^7/x^13-7/3*a^2*b^6/x^12-56/11*a^3*b^5/x^11-7*a^4*b^4/x^1
0-56/9*a^5*b^3/x^9-7/2*a^6*b^2/x^8-8/7*a^7*b/x^7-1/6*a^8/x^6

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Maxima [A]  time = 1.44093, size = 122, normalized size = 1.15 \[ -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^7,x, algorithm="maxima")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^
5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x +
1287*b^8)/x^14

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Fricas [A]  time = 0.212967, size = 122, normalized size = 1.15 \[ -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^7,x, algorithm="fricas")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^
5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x +
1287*b^8)/x^14

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Sympy [A]  time = 3.7595, size = 97, normalized size = 0.92 \[ - \frac{3003 a^{8} x^{8} + 20592 a^{7} b x^{7} + 63063 a^{6} b^{2} x^{6} + 112112 a^{5} b^{3} x^{5} + 126126 a^{4} b^{4} x^{4} + 91728 a^{3} b^{5} x^{3} + 42042 a^{2} b^{6} x^{2} + 11088 a b^{7} x + 1287 b^{8}}{18018 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**7,x)

[Out]

-(3003*a**8*x**8 + 20592*a**7*b*x**7 + 63063*a**6*b**2*x**6 + 112112*a**5*b**3*x
**5 + 126126*a**4*b**4*x**4 + 91728*a**3*b**5*x**3 + 42042*a**2*b**6*x**2 + 1108
8*a*b**7*x + 1287*b**8)/(18018*x**14)

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GIAC/XCAS [A]  time = 0.223739, size = 122, normalized size = 1.15 \[ -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^7,x, algorithm="giac")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^
5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x +
1287*b^8)/x^14