3.1586 \(\int \left (a+\frac{b}{x}\right )^8 x^{10} \, dx\)

Optimal. Leaf size=47 \[ \frac{b^2 (a x+b)^9}{9 a^3}+\frac{(a x+b)^{11}}{11 a^3}-\frac{b (a x+b)^{10}}{5 a^3} \]

[Out]

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

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Rubi [A]  time = 0.0775207, antiderivative size = 47, 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 \[ \frac{b^2 (a x+b)^9}{9 a^3}+\frac{(a x+b)^{11}}{11 a^3}-\frac{b (a x+b)^{10}}{5 a^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.093, size = 39, normalized size = 0.83 \[ \frac{b^{2} \left (a x + b\right )^{9}}{9 a^{3}} - \frac{b \left (a x + b\right )^{10}}{5 a^{3}} + \frac{\left (a x + b\right )^{11}}{11 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**10,x)

[Out]

b**2*(a*x + b)**9/(9*a**3) - b*(a*x + b)**10/(5*a**3) + (a*x + b)**11/(11*a**3)

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Mathematica [B]  time = 0.00453768, size = 102, normalized size = 2.17 \[ \frac{a^8 x^{11}}{11}+\frac{4}{5} a^7 b x^{10}+\frac{28}{9} a^6 b^2 x^9+7 a^5 b^3 x^8+10 a^4 b^4 x^7+\frac{28}{3} a^3 b^5 x^6+\frac{28}{5} a^2 b^6 x^5+2 a b^7 x^4+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^10,x]

[Out]

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

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Maple [B]  time = 0.002, size = 91, normalized size = 1.9 \[{\frac{{a}^{8}{x}^{11}}{11}}+{\frac{4\,{a}^{7}b{x}^{10}}{5}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{9}}{9}}+7\,{a}^{5}{b}^{3}{x}^{8}+10\,{a}^{4}{b}^{4}{x}^{7}+{\frac{28\,{a}^{3}{b}^{5}{x}^{6}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{5}}{5}}+2\,a{b}^{7}{x}^{4}+{\frac{{b}^{8}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^10,x)

[Out]

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

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Maxima [A]  time = 1.42572, size = 122, normalized size = 2.6 \[ \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211172, size = 122, normalized size = 2.6 \[ \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.149122, size = 102, normalized size = 2.17 \[ \frac{a^{8} x^{11}}{11} + \frac{4 a^{7} b x^{10}}{5} + \frac{28 a^{6} b^{2} x^{9}}{9} + 7 a^{5} b^{3} x^{8} + 10 a^{4} b^{4} x^{7} + \frac{28 a^{3} b^{5} x^{6}}{3} + \frac{28 a^{2} b^{6} x^{5}}{5} + 2 a b^{7} x^{4} + \frac{b^{8} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**10,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227563, size = 122, normalized size = 2.6 \[ \frac{1}{11} \, a^{8} x^{11} + \frac{4}{5} \, a^{7} b x^{10} + \frac{28}{9} \, a^{6} b^{2} x^{9} + 7 \, a^{5} b^{3} x^{8} + 10 \, a^{4} b^{4} x^{7} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{28}{5} \, a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + \frac{1}{3} \, b^{8} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^10,x, algorithm="giac")

[Out]

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