3.1581 \(\int \left (a+\frac{b}{x}\right )^8 x^{16} \, dx\)

Optimal. Leaf size=106 \[ \frac{a^8 x^{17}}{17}+\frac{1}{2} a^7 b x^{16}+\frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

[Out]

(b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 + (7
0*a^4*b^4*x^13)/13 + 4*a^5*b^3*x^14 + (28*a^6*b^2*x^15)/15 + (a^7*b*x^16)/2 + (a
^8*x^17)/17

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Rubi [A]  time = 0.134118, antiderivative size = 106, 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 \[ \frac{a^8 x^{17}}{17}+\frac{1}{2} a^7 b x^{16}+\frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^16,x]

[Out]

(b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 + (7
0*a^4*b^4*x^13)/13 + 4*a^5*b^3*x^14 + (28*a^6*b^2*x^15)/15 + (a^7*b*x^16)/2 + (a
^8*x^17)/17

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Rubi in Sympy [A]  time = 23.7752, size = 104, normalized size = 0.98 \[ \frac{a^{8} x^{17}}{17} + \frac{a^{7} b x^{16}}{2} + \frac{28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac{70 a^{4} b^{4} x^{13}}{13} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{11}}{11} + \frac{4 a b^{7} x^{10}}{5} + \frac{b^{8} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**16,x)

[Out]

a**8*x**17/17 + a**7*b*x**16/2 + 28*a**6*b**2*x**15/15 + 4*a**5*b**3*x**14 + 70*
a**4*b**4*x**13/13 + 14*a**3*b**5*x**12/3 + 28*a**2*b**6*x**11/11 + 4*a*b**7*x**
10/5 + b**8*x**9/9

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Mathematica [A]  time = 0.00551171, size = 106, normalized size = 1. \[ \frac{a^8 x^{17}}{17}+\frac{1}{2} a^7 b x^{16}+\frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^16,x]

[Out]

(b^8*x^9)/9 + (4*a*b^7*x^10)/5 + (28*a^2*b^6*x^11)/11 + (14*a^3*b^5*x^12)/3 + (7
0*a^4*b^4*x^13)/13 + 4*a^5*b^3*x^14 + (28*a^6*b^2*x^15)/15 + (a^7*b*x^16)/2 + (a
^8*x^17)/17

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Maple [A]  time = 0.004, size = 91, normalized size = 0.9 \[{\frac{{b}^{8}{x}^{9}}{9}}+{\frac{4\,a{b}^{7}{x}^{10}}{5}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{11}}{11}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{13}}{13}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{28\,{a}^{6}{b}^{2}{x}^{15}}{15}}+{\frac{{a}^{7}b{x}^{16}}{2}}+{\frac{{a}^{8}{x}^{17}}{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^16,x)

[Out]

1/9*b^8*x^9+4/5*a*b^7*x^10+28/11*a^2*b^6*x^11+14/3*a^3*b^5*x^12+70/13*a^4*b^4*x^
13+4*a^5*b^3*x^14+28/15*a^6*b^2*x^15+1/2*a^7*b*x^16+1/17*a^8*x^17

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Maxima [A]  time = 1.42684, size = 122, normalized size = 1.15 \[ \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^16,x, algorithm="maxima")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4
*b^4*x^13 + 14/3*a^3*b^5*x^12 + 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^
9

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Fricas [A]  time = 0.211439, size = 122, normalized size = 1.15 \[ \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^16,x, algorithm="fricas")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4
*b^4*x^13 + 14/3*a^3*b^5*x^12 + 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^
9

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Sympy [A]  time = 0.157539, size = 104, normalized size = 0.98 \[ \frac{a^{8} x^{17}}{17} + \frac{a^{7} b x^{16}}{2} + \frac{28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac{70 a^{4} b^{4} x^{13}}{13} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{11}}{11} + \frac{4 a b^{7} x^{10}}{5} + \frac{b^{8} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**16,x)

[Out]

a**8*x**17/17 + a**7*b*x**16/2 + 28*a**6*b**2*x**15/15 + 4*a**5*b**3*x**14 + 70*
a**4*b**4*x**13/13 + 14*a**3*b**5*x**12/3 + 28*a**2*b**6*x**11/11 + 4*a*b**7*x**
10/5 + b**8*x**9/9

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GIAC/XCAS [A]  time = 0.218663, size = 122, normalized size = 1.15 \[ \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^16,x, algorithm="giac")

[Out]

1/17*a^8*x^17 + 1/2*a^7*b*x^16 + 28/15*a^6*b^2*x^15 + 4*a^5*b^3*x^14 + 70/13*a^4
*b^4*x^13 + 14/3*a^3*b^5*x^12 + 28/11*a^2*b^6*x^11 + 4/5*a*b^7*x^10 + 1/9*b^8*x^
9