3.1587 \(\int \left (a+\frac{b}{x}\right )^8 x^9 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a x+b)^{10}}{10 a^2}-\frac{b (a x+b)^9}{9 a^2} \]

[Out]

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

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Rubi [A]  time = 0.042959, antiderivative size = 30, 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 \[ \frac{(a x+b)^{10}}{10 a^2}-\frac{b (a x+b)^9}{9 a^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1946, size = 24, normalized size = 0.8 \[ - \frac{b \left (a x + b\right )^{9}}{9 a^{2}} + \frac{\left (a x + b\right )^{10}}{10 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**9,x)

[Out]

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

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Mathematica [B]  time = 0.00431785, size = 104, normalized size = 3.47 \[ \frac{a^8 x^{10}}{10}+\frac{8}{9} a^7 b x^9+\frac{7}{2} a^6 b^2 x^8+8 a^5 b^3 x^7+\frac{35}{3} a^4 b^4 x^6+\frac{56}{5} a^3 b^5 x^5+7 a^2 b^6 x^4+\frac{8}{3} a b^7 x^3+\frac{b^8 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^9,x]

[Out]

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

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Maple [B]  time = 0.002, size = 91, normalized size = 3. \[{\frac{{a}^{8}{x}^{10}}{10}}+{\frac{8\,{a}^{7}b{x}^{9}}{9}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{8}}{2}}+8\,{a}^{5}{b}^{3}{x}^{7}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{5}}{5}}+7\,{a}^{2}{b}^{6}{x}^{4}+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{x}^{2}{b}^{8}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^9,x)

[Out]

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

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Maxima [A]  time = 1.44819, size = 122, normalized size = 4.07 \[ \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^9,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211513, size = 122, normalized size = 4.07 \[ \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^9,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.1548, size = 104, normalized size = 3.47 \[ \frac{a^{8} x^{10}}{10} + \frac{8 a^{7} b x^{9}}{9} + \frac{7 a^{6} b^{2} x^{8}}{2} + 8 a^{5} b^{3} x^{7} + \frac{35 a^{4} b^{4} x^{6}}{3} + \frac{56 a^{3} b^{5} x^{5}}{5} + 7 a^{2} b^{6} x^{4} + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**9,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221456, size = 122, normalized size = 4.07 \[ \frac{1}{10} \, a^{8} x^{10} + \frac{8}{9} \, a^{7} b x^{9} + \frac{7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac{8}{3} \, a b^{7} x^{3} + \frac{1}{2} \, b^{8} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^9,x, algorithm="giac")

[Out]

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