3.1604 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^8} \, dx\)

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

[Out]

-b^8/(15*x^15) - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5)/(3*x
^12) - (70*a^4*b^4)/(11*x^11) - (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2)/(9*x^9) - (
a^7*b)/x^8 - a^8/(7*x^7)

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Rubi [A]  time = 0.113035, 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 \[ -\frac{a^8}{7 x^7}-\frac{a^7 b}{x^8}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{14 a^3 b^5}{3 x^{12}}-\frac{28 a^2 b^6}{13 x^{13}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^8,x]

[Out]

-b^8/(15*x^15) - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5)/(3*x
^12) - (70*a^4*b^4)/(11*x^11) - (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2)/(9*x^9) - (
a^7*b)/x^8 - a^8/(7*x^7)

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Rubi in Sympy [A]  time = 20.717, size = 105, normalized size = 0.99 \[ - \frac{a^{8}}{7 x^{7}} - \frac{a^{7} b}{x^{8}} - \frac{28 a^{6} b^{2}}{9 x^{9}} - \frac{28 a^{5} b^{3}}{5 x^{10}} - \frac{70 a^{4} b^{4}}{11 x^{11}} - \frac{14 a^{3} b^{5}}{3 x^{12}} - \frac{28 a^{2} b^{6}}{13 x^{13}} - \frac{4 a b^{7}}{7 x^{14}} - \frac{b^{8}}{15 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**8/(7*x**7) - a**7*b/x**8 - 28*a**6*b**2/(9*x**9) - 28*a**5*b**3/(5*x**10) -
70*a**4*b**4/(11*x**11) - 14*a**3*b**5/(3*x**12) - 28*a**2*b**6/(13*x**13) - 4*a
*b**7/(7*x**14) - b**8/(15*x**15)

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Mathematica [A]  time = 0.0111722, size = 106, normalized size = 1. \[ -\frac{a^8}{7 x^7}-\frac{a^7 b}{x^8}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{14 a^3 b^5}{3 x^{12}}-\frac{28 a^2 b^6}{13 x^{13}}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^8,x]

[Out]

-b^8/(15*x^15) - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5)/(3*x
^12) - (70*a^4*b^4)/(11*x^11) - (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2)/(9*x^9) - (
a^7*b)/x^8 - a^8/(7*x^7)

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Maple [A]  time = 0.009, size = 91, normalized size = 0.9 \[ -{\frac{{b}^{8}}{15\,{x}^{15}}}-{\frac{4\,a{b}^{7}}{7\,{x}^{14}}}-{\frac{28\,{a}^{2}{b}^{6}}{13\,{x}^{13}}}-{\frac{14\,{a}^{3}{b}^{5}}{3\,{x}^{12}}}-{\frac{70\,{a}^{4}{b}^{4}}{11\,{x}^{11}}}-{\frac{28\,{a}^{5}{b}^{3}}{5\,{x}^{10}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,{x}^{9}}}-{\frac{{a}^{7}b}{{x}^{8}}}-{\frac{{a}^{8}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^8,x)

[Out]

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

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Maxima [A]  time = 1.44572, size = 122, normalized size = 1.15 \[ -\frac{6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x
^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x
+ 3003*b^8)/x^15

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Fricas [A]  time = 0.211984, size = 122, normalized size = 1.15 \[ -\frac{6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x
^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x
+ 3003*b^8)/x^15

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Sympy [A]  time = 3.89265, size = 97, normalized size = 0.92 \[ - \frac{6435 a^{8} x^{8} + 45045 a^{7} b x^{7} + 140140 a^{6} b^{2} x^{6} + 252252 a^{5} b^{3} x^{5} + 286650 a^{4} b^{4} x^{4} + 210210 a^{3} b^{5} x^{3} + 97020 a^{2} b^{6} x^{2} + 25740 a b^{7} x + 3003 b^{8}}{45045 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**8,x)

[Out]

-(6435*a**8*x**8 + 45045*a**7*b*x**7 + 140140*a**6*b**2*x**6 + 252252*a**5*b**3*
x**5 + 286650*a**4*b**4*x**4 + 210210*a**3*b**5*x**3 + 97020*a**2*b**6*x**2 + 25
740*a*b**7*x + 3003*b**8)/(45045*x**15)

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GIAC/XCAS [A]  time = 0.22522, size = 122, normalized size = 1.15 \[ -\frac{6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x
^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x
+ 3003*b^8)/x^15