3.1597 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x} \, dx\)

Optimal. Leaf size=100 \[ a^8 \log (x)-\frac{8 a^7 b}{x}-\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) -
(35*a^4*b^4)/(2*x^4) - (56*a^5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x +
a^8*Log[x]

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Rubi [A]  time = 0.108362, antiderivative size = 100, 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 \[ a^8 \log (x)-\frac{8 a^7 b}{x}-\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x,x]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) -
(35*a^4*b^4)/(2*x^4) - (56*a^5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x +
a^8*Log[x]

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Rubi in Sympy [A]  time = 20.3526, size = 100, normalized size = 1. \[ a^{8} \log{\left (x \right )} - \frac{8 a^{7} b}{x} - \frac{14 a^{6} b^{2}}{x^{2}} - \frac{56 a^{5} b^{3}}{3 x^{3}} - \frac{35 a^{4} b^{4}}{2 x^{4}} - \frac{56 a^{3} b^{5}}{5 x^{5}} - \frac{14 a^{2} b^{6}}{3 x^{6}} - \frac{8 a b^{7}}{7 x^{7}} - \frac{b^{8}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8/x,x)

[Out]

a**8*log(x) - 8*a**7*b/x - 14*a**6*b**2/x**2 - 56*a**5*b**3/(3*x**3) - 35*a**4*b
**4/(2*x**4) - 56*a**3*b**5/(5*x**5) - 14*a**2*b**6/(3*x**6) - 8*a*b**7/(7*x**7)
- b**8/(8*x**8)

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Mathematica [A]  time = 0.00770999, size = 100, normalized size = 1. \[ a^8 \log (x)-\frac{8 a^7 b}{x}-\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x,x]

[Out]

-b^8/(8*x^8) - (8*a*b^7)/(7*x^7) - (14*a^2*b^6)/(3*x^6) - (56*a^3*b^5)/(5*x^5) -
(35*a^4*b^4)/(2*x^4) - (56*a^5*b^3)/(3*x^3) - (14*a^6*b^2)/x^2 - (8*a^7*b)/x +
a^8*Log[x]

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Maple [A]  time = 0.012, size = 89, normalized size = 0.9 \[ -{\frac{{b}^{8}}{8\,{x}^{8}}}-{\frac{8\,a{b}^{7}}{7\,{x}^{7}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,{x}^{5}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,{x}^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}-8\,{\frac{{a}^{7}b}{x}}+{a}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x,x)

[Out]

-1/8*b^8/x^8-8/7*a*b^7/x^7-14/3*a^2*b^6/x^6-56/5*a^3*b^5/x^5-35/2*a^4*b^4/x^4-56
/3*a^5*b^3/x^3-14*a^6*b^2/x^2-8*a^7*b/x+a^8*ln(x)

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Maxima [A]  time = 1.43028, size = 120, normalized size = 1.2 \[ a^{8} \log \left (x\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^8*log(x) - 1/840*(6720*a^7*b*x^7 + 11760*a^6*b^2*x^6 + 15680*a^5*b^3*x^5 + 147
00*a^4*b^4*x^4 + 9408*a^3*b^5*x^3 + 3920*a^2*b^6*x^2 + 960*a*b^7*x + 105*b^8)/x^
8

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Fricas [A]  time = 0.219911, size = 124, normalized size = 1.24 \[ \frac{840 \, a^{8} x^{8} \log \left (x\right ) - 6720 \, a^{7} b x^{7} - 11760 \, a^{6} b^{2} x^{6} - 15680 \, a^{5} b^{3} x^{5} - 14700 \, a^{4} b^{4} x^{4} - 9408 \, a^{3} b^{5} x^{3} - 3920 \, a^{2} b^{6} x^{2} - 960 \, a b^{7} x - 105 \, b^{8}}{840 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(840*a^8*x^8*log(x) - 6720*a^7*b*x^7 - 11760*a^6*b^2*x^6 - 15680*a^5*b^3*x
^5 - 14700*a^4*b^4*x^4 - 9408*a^3*b^5*x^3 - 3920*a^2*b^6*x^2 - 960*a*b^7*x - 105
*b^8)/x^8

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Sympy [A]  time = 2.81548, size = 94, normalized size = 0.94 \[ a^{8} \log{\left (x \right )} - \frac{6720 a^{7} b x^{7} + 11760 a^{6} b^{2} x^{6} + 15680 a^{5} b^{3} x^{5} + 14700 a^{4} b^{4} x^{4} + 9408 a^{3} b^{5} x^{3} + 3920 a^{2} b^{6} x^{2} + 960 a b^{7} x + 105 b^{8}}{840 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x,x)

[Out]

a**8*log(x) - (6720*a**7*b*x**7 + 11760*a**6*b**2*x**6 + 15680*a**5*b**3*x**5 +
14700*a**4*b**4*x**4 + 9408*a**3*b**5*x**3 + 3920*a**2*b**6*x**2 + 960*a*b**7*x
+ 105*b**8)/(840*x**8)

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GIAC/XCAS [A]  time = 0.223878, size = 122, normalized size = 1.22 \[ a^{8}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^8*ln(abs(x)) - 1/840*(6720*a^7*b*x^7 + 11760*a^6*b^2*x^6 + 15680*a^5*b^3*x^5 +
14700*a^4*b^4*x^4 + 9408*a^3*b^5*x^3 + 3920*a^2*b^6*x^2 + 960*a*b^7*x + 105*b^8
)/x^8