3.1584 \(\int \left (a+\frac{b}{x}\right )^8 x^{12} \, dx\)

Optimal. Leaf size=81 \[ \frac{b^4 (a x+b)^9}{9 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{6 b^2 (a x+b)^{11}}{11 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]

[Out]

(b^4*(b + a*x)^9)/(9*a^5) - (2*b^3*(b + a*x)^10)/(5*a^5) + (6*b^2*(b + a*x)^11)/
(11*a^5) - (b*(b + a*x)^12)/(3*a^5) + (b + a*x)^13/(13*a^5)

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Rubi [A]  time = 0.109553, antiderivative size = 81, 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 \[ \frac{b^4 (a x+b)^9}{9 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{6 b^2 (a x+b)^{11}}{11 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^12,x]

[Out]

(b^4*(b + a*x)^9)/(9*a^5) - (2*b^3*(b + a*x)^10)/(5*a^5) + (6*b^2*(b + a*x)^11)/
(11*a^5) - (b*(b + a*x)^12)/(3*a^5) + (b + a*x)^13/(13*a^5)

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Rubi in Sympy [A]  time = 21.0719, size = 73, normalized size = 0.9 \[ \frac{b^{4} \left (a x + b\right )^{9}}{9 a^{5}} - \frac{2 b^{3} \left (a x + b\right )^{10}}{5 a^{5}} + \frac{6 b^{2} \left (a x + b\right )^{11}}{11 a^{5}} - \frac{b \left (a x + b\right )^{12}}{3 a^{5}} + \frac{\left (a x + b\right )^{13}}{13 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**12,x)

[Out]

b**4*(a*x + b)**9/(9*a**5) - 2*b**3*(a*x + b)**10/(5*a**5) + 6*b**2*(a*x + b)**1
1/(11*a**5) - b*(a*x + b)**12/(3*a**5) + (a*x + b)**13/(13*a**5)

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Mathematica [A]  time = 0.00551555, size = 104, normalized size = 1.28 \[ \frac{a^8 x^{13}}{13}+\frac{2}{3} a^7 b x^{12}+\frac{28}{11} a^6 b^2 x^{11}+\frac{28}{5} a^5 b^3 x^{10}+\frac{70}{9} a^4 b^4 x^9+7 a^3 b^5 x^8+4 a^2 b^6 x^7+\frac{4}{3} a b^7 x^6+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^12,x]

[Out]

(b^8*x^5)/5 + (4*a*b^7*x^6)/3 + 4*a^2*b^6*x^7 + 7*a^3*b^5*x^8 + (70*a^4*b^4*x^9)
/9 + (28*a^5*b^3*x^10)/5 + (28*a^6*b^2*x^11)/11 + (2*a^7*b*x^12)/3 + (a^8*x^13)/
13

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Maple [A]  time = 0.003, size = 91, normalized size = 1.1 \[{\frac{{a}^{8}{x}^{13}}{13}}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{11}}{11}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+7\,{a}^{3}{b}^{5}{x}^{8}+4\,{a}^{2}{b}^{6}{x}^{7}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^12,x)

[Out]

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

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Maxima [A]  time = 1.44445, size = 122, normalized size = 1.51 \[ \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^12,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212989, size = 122, normalized size = 1.51 \[ \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^12,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.154342, size = 104, normalized size = 1.28 \[ \frac{a^{8} x^{13}}{13} + \frac{2 a^{7} b x^{12}}{3} + \frac{28 a^{6} b^{2} x^{11}}{11} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{70 a^{4} b^{4} x^{9}}{9} + 7 a^{3} b^{5} x^{8} + 4 a^{2} b^{6} x^{7} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**12,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.225425, size = 122, normalized size = 1.51 \[ \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^12,x, algorithm="giac")

[Out]

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