3.1593 \(\int \left (a+\frac{b}{x}\right )^8 x^3 \, dx\)

Optimal. Leaf size=95 \[ \frac{a^8 x^4}{4}+\frac{8}{3} a^7 b x^3+14 a^6 b^2 x^2+56 a^5 b^3 x+70 a^4 b^4 \log (x)-\frac{56 a^3 b^5}{x}-\frac{14 a^2 b^6}{x^2}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

[Out]

-b^8/(4*x^4) - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^
3*x + 14*a^6*b^2*x^2 + (8*a^7*b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.110693, 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 \[ \frac{a^8 x^4}{4}+\frac{8}{3} a^7 b x^3+14 a^6 b^2 x^2+56 a^5 b^3 x+70 a^4 b^4 \log (x)-\frac{56 a^3 b^5}{x}-\frac{14 a^2 b^6}{x^2}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^3,x]

[Out]

-b^8/(4*x^4) - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^
3*x + 14*a^6*b^2*x^2 + (8*a^7*b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} x^{4}}{4} + \frac{8 a^{7} b x^{3}}{3} + 28 a^{6} b^{2} \int x\, dx + 56 a^{5} b^{3} x + 70 a^{4} b^{4} \log{\left (x \right )} - \frac{56 a^{3} b^{5}}{x} - \frac{14 a^{2} b^{6}}{x^{2}} - \frac{8 a b^{7}}{3 x^{3}} - \frac{b^{8}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**3,x)

[Out]

a**8*x**4/4 + 8*a**7*b*x**3/3 + 28*a**6*b**2*Integral(x, x) + 56*a**5*b**3*x + 7
0*a**4*b**4*log(x) - 56*a**3*b**5/x - 14*a**2*b**6/x**2 - 8*a*b**7/(3*x**3) - b*
*8/(4*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00826132, size = 95, normalized size = 1. \[ \frac{a^8 x^4}{4}+\frac{8}{3} a^7 b x^3+14 a^6 b^2 x^2+56 a^5 b^3 x+70 a^4 b^4 \log (x)-\frac{56 a^3 b^5}{x}-\frac{14 a^2 b^6}{x^2}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^3,x]

[Out]

-b^8/(4*x^4) - (8*a*b^7)/(3*x^3) - (14*a^2*b^6)/x^2 - (56*a^3*b^5)/x + 56*a^5*b^
3*x + 14*a^6*b^2*x^2 + (8*a^7*b*x^3)/3 + (a^8*x^4)/4 + 70*a^4*b^4*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{8}}{4\,{x}^{4}}}-{\frac{8\,a{b}^{7}}{3\,{x}^{3}}}-14\,{\frac{{a}^{2}{b}^{6}}{{x}^{2}}}-56\,{\frac{{a}^{3}{b}^{5}}{x}}+56\,{a}^{5}{b}^{3}x+14\,{a}^{6}{b}^{2}{x}^{2}+{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{{a}^{8}{x}^{4}}{4}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^3,x)

[Out]

-1/4*b^8/x^4-8/3*a*b^7/x^3-14*a^2*b^6/x^2-56*a^3*b^5/x+56*a^5*b^3*x+14*a^6*b^2*x
^2+8/3*a^7*b*x^3+1/4*a^8*x^4+70*a^4*b^4*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.41908, size = 119, normalized size = 1.25 \[ \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left (x\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^3,x, algorithm="maxima")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*log(x)
- 1/12*(672*a^3*b^5*x^3 + 168*a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.219826, size = 124, normalized size = 1.31 \[ \frac{3 \, a^{8} x^{8} + 32 \, a^{7} b x^{7} + 168 \, a^{6} b^{2} x^{6} + 672 \, a^{5} b^{3} x^{5} + 840 \, a^{4} b^{4} x^{4} \log \left (x\right ) - 672 \, a^{3} b^{5} x^{3} - 168 \, a^{2} b^{6} x^{2} - 32 \, a b^{7} x - 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^3,x, algorithm="fricas")

[Out]

1/12*(3*a^8*x^8 + 32*a^7*b*x^7 + 168*a^6*b^2*x^6 + 672*a^5*b^3*x^5 + 840*a^4*b^4
*x^4*log(x) - 672*a^3*b^5*x^3 - 168*a^2*b^6*x^2 - 32*a*b^7*x - 3*b^8)/x^4

_______________________________________________________________________________________

Sympy [A]  time = 1.87063, size = 95, normalized size = 1. \[ \frac{a^{8} x^{4}}{4} + \frac{8 a^{7} b x^{3}}{3} + 14 a^{6} b^{2} x^{2} + 56 a^{5} b^{3} x + 70 a^{4} b^{4} \log{\left (x \right )} - \frac{672 a^{3} b^{5} x^{3} + 168 a^{2} b^{6} x^{2} + 32 a b^{7} x + 3 b^{8}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**3,x)

[Out]

a**8*x**4/4 + 8*a**7*b*x**3/3 + 14*a**6*b**2*x**2 + 56*a**5*b**3*x + 70*a**4*b**
4*log(x) - (672*a**3*b**5*x**3 + 168*a**2*b**6*x**2 + 32*a*b**7*x + 3*b**8)/(12*
x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225739, size = 120, normalized size = 1.26 \[ \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^3,x, algorithm="giac")

[Out]

1/4*a^8*x^4 + 8/3*a^7*b*x^3 + 14*a^6*b^2*x^2 + 56*a^5*b^3*x + 70*a^4*b^4*ln(abs(
x)) - 1/12*(672*a^3*b^5*x^3 + 168*a^2*b^6*x^2 + 32*a*b^7*x + 3*b^8)/x^4