3.1591 \(\int \left (a+\frac{b}{x}\right )^8 x^5 \, dx\)

Optimal. Leaf size=95 \[ \frac{a^8 x^6}{6}+\frac{8}{5} a^7 b x^5+7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3
+ 7*a^6*b^2*x^4 + (8*a^7*b*x^5)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.109227, antiderivative size = 95, 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 \[ \frac{a^8 x^6}{6}+\frac{8}{5} a^7 b x^5+7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^5,x]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3
+ 7*a^6*b^2*x^4 + (8*a^7*b*x^5)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} x^{6}}{6} + \frac{8 a^{7} b x^{5}}{5} + 7 a^{6} b^{2} x^{4} + \frac{56 a^{5} b^{3} x^{3}}{3} + 70 a^{4} b^{4} \int x\, dx + 56 a^{3} b^{5} x + 28 a^{2} b^{6} \log{\left (x \right )} - \frac{8 a b^{7}}{x} - \frac{b^{8}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**8*x**6/6 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**4 + 56*a**5*b**3*x**3/3 + 70*a**4
*b**4*Integral(x, x) + 56*a**3*b**5*x + 28*a**2*b**6*log(x) - 8*a*b**7/x - b**8/
(2*x**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00818389, size = 95, normalized size = 1. \[ \frac{a^8 x^6}{6}+\frac{8}{5} a^7 b x^5+7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^5,x]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3
+ 7*a^6*b^2*x^4 + (8*a^7*b*x^5)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{8}}{2\,{x}^{2}}}-8\,{\frac{a{b}^{7}}{x}}+56\,{a}^{3}{b}^{5}x+35\,{a}^{4}{b}^{4}{x}^{2}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+7\,{a}^{6}{b}^{2}{x}^{4}+{\frac{8\,{a}^{7}b{x}^{5}}{5}}+{\frac{{a}^{8}{x}^{6}}{6}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^5,x)

[Out]

-1/2*b^8/x^2-8*a*b^7/x+56*a^3*b^5*x+35*a^4*b^4*x^2+56/3*a^5*b^3*x^3+7*a^6*b^2*x^
4+8/5*a^7*b*x^5+1/6*a^8*x^6+28*a^2*b^6*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.43399, size = 116, normalized size = 1.22 \[ \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left (x\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2
+ 56*a^3*b^5*x + 28*a^2*b^6*log(x) - 1/2*(16*a*b^7*x + b^8)/x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.219777, size = 124, normalized size = 1.31 \[ \frac{5 \, a^{8} x^{8} + 48 \, a^{7} b x^{7} + 210 \, a^{6} b^{2} x^{6} + 560 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 1680 \, a^{3} b^{5} x^{3} + 840 \, a^{2} b^{6} x^{2} \log \left (x\right ) - 240 \, a b^{7} x - 15 \, b^{8}}{30 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(5*a^8*x^8 + 48*a^7*b*x^7 + 210*a^6*b^2*x^6 + 560*a^5*b^3*x^5 + 1050*a^4*b^
4*x^4 + 1680*a^3*b^5*x^3 + 840*a^2*b^6*x^2*log(x) - 240*a*b^7*x - 15*b^8)/x^2

_______________________________________________________________________________________

Sympy [A]  time = 1.53222, size = 95, normalized size = 1. \[ \frac{a^{8} x^{6}}{6} + \frac{8 a^{7} b x^{5}}{5} + 7 a^{6} b^{2} x^{4} + \frac{56 a^{5} b^{3} x^{3}}{3} + 35 a^{4} b^{4} x^{2} + 56 a^{3} b^{5} x + 28 a^{2} b^{6} \log{\left (x \right )} - \frac{16 a b^{7} x + b^{8}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**5,x)

[Out]

a**8*x**6/6 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**4 + 56*a**5*b**3*x**3/3 + 35*a**4
*b**4*x**2 + 56*a**3*b**5*x + 28*a**2*b**6*log(x) - (16*a*b**7*x + b**8)/(2*x**2
)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22386, size = 117, normalized size = 1.23 \[ \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6}{\rm ln}\left ({\left | x \right |}\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2
+ 56*a^3*b^5*x + 28*a^2*b^6*ln(abs(x)) - 1/2*(16*a*b^7*x + b^8)/x^2