3.1599 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{a (a x+b)^9}{90 b^2 x^9}-\frac{(a x+b)^9}{10 b x^{10}} \]

[Out]

-(b + a*x)^9/(10*b*x^10) + (a*(b + a*x)^9)/(90*b^2*x^9)

_______________________________________________________________________________________

Rubi [A]  time = 0.038869, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a x+b)^9}{90 b^2 x^9}-\frac{(a x+b)^9}{10 b x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^3,x]

[Out]

-(b + a*x)^9/(10*b*x^10) + (a*(b + a*x)^9)/(90*b^2*x^9)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.43205, size = 29, normalized size = 0.81 \[ \frac{a \left (a x + b\right )^{9}}{90 b^{2} x^{9}} - \frac{\left (a x + b\right )^{9}}{10 b x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(a*x + b)**9/(90*b**2*x**9) - (a*x + b)**9/(10*b*x**10)

_______________________________________________________________________________________

Mathematica [B]  time = 0.00734905, size = 104, normalized size = 2.89 \[ -\frac{a^8}{2 x^2}-\frac{8 a^7 b}{3 x^3}-\frac{7 a^6 b^2}{x^4}-\frac{56 a^5 b^3}{5 x^5}-\frac{35 a^4 b^4}{3 x^6}-\frac{8 a^3 b^5}{x^7}-\frac{7 a^2 b^6}{2 x^8}-\frac{8 a b^7}{9 x^9}-\frac{b^8}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^3,x]

[Out]

-b^8/(10*x^10) - (8*a*b^7)/(9*x^9) - (7*a^2*b^6)/(2*x^8) - (8*a^3*b^5)/x^7 - (35
*a^4*b^4)/(3*x^6) - (56*a^5*b^3)/(5*x^5) - (7*a^6*b^2)/x^4 - (8*a^7*b)/(3*x^3) -
a^8/(2*x^2)

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 91, normalized size = 2.5 \[ -{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-7\,{\frac{{a}^{6}{b}^{2}}{{x}^{4}}}-{\frac{{b}^{8}}{10\,{x}^{10}}}-{\frac{7\,{a}^{2}{b}^{6}}{2\,{x}^{8}}}-{\frac{8\,a{b}^{7}}{9\,{x}^{9}}}-{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}-{\frac{{a}^{8}}{2\,{x}^{2}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,{x}^{5}}}-8\,{\frac{{a}^{3}{b}^{5}}{{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^3,x)

[Out]

-35/3*a^4*b^4/x^6-7*a^6*b^2/x^4-1/10*b^8/x^10-7/2*a^2*b^6/x^8-8/9*a*b^7/x^9-8/3*
a^7*b/x^3-1/2*a^8/x^2-56/5*a^5*b^3/x^5-8*a^3*b^5/x^7

_______________________________________________________________________________________

Maxima [A]  time = 1.44484, size = 122, normalized size = 3.39 \[ -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^
4*b^4*x^4 + 720*a^3*b^5*x^3 + 315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10

_______________________________________________________________________________________

Fricas [A]  time = 0.216041, size = 122, normalized size = 3.39 \[ -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^
4*b^4*x^4 + 720*a^3*b^5*x^3 + 315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10

_______________________________________________________________________________________

Sympy [A]  time = 3.14696, size = 97, normalized size = 2.69 \[ - \frac{45 a^{8} x^{8} + 240 a^{7} b x^{7} + 630 a^{6} b^{2} x^{6} + 1008 a^{5} b^{3} x^{5} + 1050 a^{4} b^{4} x^{4} + 720 a^{3} b^{5} x^{3} + 315 a^{2} b^{6} x^{2} + 80 a b^{7} x + 9 b^{8}}{90 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**3,x)

[Out]

-(45*a**8*x**8 + 240*a**7*b*x**7 + 630*a**6*b**2*x**6 + 1008*a**5*b**3*x**5 + 10
50*a**4*b**4*x**4 + 720*a**3*b**5*x**3 + 315*a**2*b**6*x**2 + 80*a*b**7*x + 9*b*
*8)/(90*x**10)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22792, size = 122, normalized size = 3.39 \[ -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^
4*b^4*x^4 + 720*a^3*b^5*x^3 + 315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10