3.1596 \(\int \left (a+\frac{b}{x}\right )^8 \, dx\)

Optimal. Leaf size=94 \[ a^8 x+8 a^7 b \log (x)-\frac{28 a^6 b^2}{x}-\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70
*a^4*b^4)/(3*x^3) - (28*a^5*b^3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.105179, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ a^8 x+8 a^7 b \log (x)-\frac{28 a^6 b^2}{x}-\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8,x]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70
*a^4*b^4)/(3*x^3) - (28*a^5*b^3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 8 a^{7} b \log{\left (x \right )} - \frac{28 a^{6} b^{2}}{x} - \frac{28 a^{5} b^{3}}{x^{2}} - \frac{70 a^{4} b^{4}}{3 x^{3}} - \frac{14 a^{3} b^{5}}{x^{4}} - \frac{28 a^{2} b^{6}}{5 x^{5}} - \frac{4 a b^{7}}{3 x^{6}} - \frac{b^{8}}{7 x^{7}} + \int a^{8}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8,x)

[Out]

8*a**7*b*log(x) - 28*a**6*b**2/x - 28*a**5*b**3/x**2 - 70*a**4*b**4/(3*x**3) - 1
4*a**3*b**5/x**4 - 28*a**2*b**6/(5*x**5) - 4*a*b**7/(3*x**6) - b**8/(7*x**7) + I
ntegral(a**8, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00891505, size = 94, normalized size = 1. \[ a^8 x+8 a^7 b \log (x)-\frac{28 a^6 b^2}{x}-\frac{28 a^5 b^3}{x^2}-\frac{70 a^4 b^4}{3 x^3}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{5 x^5}-\frac{4 a b^7}{3 x^6}-\frac{b^8}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8,x]

[Out]

-b^8/(7*x^7) - (4*a*b^7)/(3*x^6) - (28*a^2*b^6)/(5*x^5) - (14*a^3*b^5)/x^4 - (70
*a^4*b^4)/(3*x^3) - (28*a^5*b^3)/x^2 - (28*a^6*b^2)/x + a^8*x + 8*a^7*b*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 87, normalized size = 0.9 \[ -{\frac{{b}^{8}}{7\,{x}^{7}}}-{\frac{4\,a{b}^{7}}{3\,{x}^{6}}}-{\frac{28\,{a}^{2}{b}^{6}}{5\,{x}^{5}}}-14\,{\frac{{a}^{3}{b}^{5}}{{x}^{4}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}-28\,{\frac{{a}^{5}{b}^{3}}{{x}^{2}}}-28\,{\frac{{a}^{6}{b}^{2}}{x}}+{a}^{8}x+8\,{a}^{7}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8,x)

[Out]

-1/7*b^8/x^7-4/3*a*b^7/x^6-28/5*a^2*b^6/x^5-14*a^3*b^5/x^4-70/3*a^4*b^4/x^3-28*a
^5*b^3/x^2-28*a^6*b^2/x+a^8*x+8*a^7*b*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.44189, size = 117, normalized size = 1.24 \[ a^{8} x + 8 \, a^{7} b \log \left (x\right ) - \frac{2940 \, a^{6} b^{2} x^{6} + 2940 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 1470 \, a^{3} b^{5} x^{3} + 588 \, a^{2} b^{6} x^{2} + 140 \, a b^{7} x + 15 \, b^{8}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^8*x + 8*a^7*b*log(x) - 1/105*(2940*a^6*b^2*x^6 + 2940*a^5*b^3*x^5 + 2450*a^4*b
^4*x^4 + 1470*a^3*b^5*x^3 + 588*a^2*b^6*x^2 + 140*a*b^7*x + 15*b^8)/x^7

_______________________________________________________________________________________

Fricas [A]  time = 0.218617, size = 124, normalized size = 1.32 \[ \frac{105 \, a^{8} x^{8} + 840 \, a^{7} b x^{7} \log \left (x\right ) - 2940 \, a^{6} b^{2} x^{6} - 2940 \, a^{5} b^{3} x^{5} - 2450 \, a^{4} b^{4} x^{4} - 1470 \, a^{3} b^{5} x^{3} - 588 \, a^{2} b^{6} x^{2} - 140 \, a b^{7} x - 15 \, b^{8}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(105*a^8*x^8 + 840*a^7*b*x^7*log(x) - 2940*a^6*b^2*x^6 - 2940*a^5*b^3*x^5
- 2450*a^4*b^4*x^4 - 1470*a^3*b^5*x^3 - 588*a^2*b^6*x^2 - 140*a*b^7*x - 15*b^8)/
x^7

_______________________________________________________________________________________

Sympy [A]  time = 2.62155, size = 92, normalized size = 0.98 \[ a^{8} x + 8 a^{7} b \log{\left (x \right )} - \frac{2940 a^{6} b^{2} x^{6} + 2940 a^{5} b^{3} x^{5} + 2450 a^{4} b^{4} x^{4} + 1470 a^{3} b^{5} x^{3} + 588 a^{2} b^{6} x^{2} + 140 a b^{7} x + 15 b^{8}}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8,x)

[Out]

a**8*x + 8*a**7*b*log(x) - (2940*a**6*b**2*x**6 + 2940*a**5*b**3*x**5 + 2450*a**
4*b**4*x**4 + 1470*a**3*b**5*x**3 + 588*a**2*b**6*x**2 + 140*a*b**7*x + 15*b**8)
/(105*x**7)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222334, size = 119, normalized size = 1.27 \[ a^{8} x + 8 \, a^{7} b{\rm ln}\left ({\left | x \right |}\right ) - \frac{2940 \, a^{6} b^{2} x^{6} + 2940 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 1470 \, a^{3} b^{5} x^{3} + 588 \, a^{2} b^{6} x^{2} + 140 \, a b^{7} x + 15 \, b^{8}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^8*x + 8*a^7*b*ln(abs(x)) - 1/105*(2940*a^6*b^2*x^6 + 2940*a^5*b^3*x^5 + 2450*a
^4*b^4*x^4 + 1470*a^3*b^5*x^3 + 588*a^2*b^6*x^2 + 140*a*b^7*x + 15*b^8)/x^7