3.1589 \(\int \left (a+\frac{b}{x}\right )^8 x^7 \, dx\)

Optimal. Leaf size=98 \[ \frac{a^8 x^8}{8}+\frac{8}{7} a^7 b x^7+\frac{14}{3} a^6 b^2 x^6+\frac{56}{5} a^5 b^3 x^5+\frac{35}{2} a^4 b^4 x^4+\frac{56}{3} a^3 b^5 x^3+14 a^2 b^6 x^2+8 a b^7 x+b^8 \log (x) \]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b
^3*x^5)/5 + (14*a^6*b^2*x^6)/3 + (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*Log[x]

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Rubi [A]  time = 0.0930235, antiderivative size = 98, 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 \[ \frac{a^8 x^8}{8}+\frac{8}{7} a^7 b x^7+\frac{14}{3} a^6 b^2 x^6+\frac{56}{5} a^5 b^3 x^5+\frac{35}{2} a^4 b^4 x^4+\frac{56}{3} a^3 b^5 x^3+14 a^2 b^6 x^2+8 a b^7 x+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^7,x]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b
^3*x^5)/5 + (14*a^6*b^2*x^6)/3 + (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} x^{8}}{8} + \frac{8 a^{7} b x^{7}}{7} + \frac{14 a^{6} b^{2} x^{6}}{3} + \frac{56 a^{5} b^{3} x^{5}}{5} + \frac{35 a^{4} b^{4} x^{4}}{2} + \frac{56 a^{3} b^{5} x^{3}}{3} + 28 a^{2} b^{6} \int x\, dx + 8 a b^{7} x + b^{8} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**7,x)

[Out]

a**8*x**8/8 + 8*a**7*b*x**7/7 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**5/5 + 35*a
**4*b**4*x**4/2 + 56*a**3*b**5*x**3/3 + 28*a**2*b**6*Integral(x, x) + 8*a*b**7*x
+ b**8*log(x)

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Mathematica [A]  time = 0.00603072, size = 98, normalized size = 1. \[ \frac{a^8 x^8}{8}+\frac{8}{7} a^7 b x^7+\frac{14}{3} a^6 b^2 x^6+\frac{56}{5} a^5 b^3 x^5+\frac{35}{2} a^4 b^4 x^4+\frac{56}{3} a^3 b^5 x^3+14 a^2 b^6 x^2+8 a b^7 x+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^7,x]

[Out]

8*a*b^7*x + 14*a^2*b^6*x^2 + (56*a^3*b^5*x^3)/3 + (35*a^4*b^4*x^4)/2 + (56*a^5*b
^3*x^5)/5 + (14*a^6*b^2*x^6)/3 + (8*a^7*b*x^7)/7 + (a^8*x^8)/8 + b^8*Log[x]

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Maple [A]  time = 0.004, size = 87, normalized size = 0.9 \[ 8\,a{b}^{7}x+14\,{a}^{2}{b}^{6}{x}^{2}+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{4}}{2}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{5}}{5}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{6}}{3}}+{\frac{8\,{a}^{7}b{x}^{7}}{7}}+{\frac{{a}^{8}{x}^{8}}{8}}+{b}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^7,x)

[Out]

8*a*b^7*x+14*a^2*b^6*x^2+56/3*a^3*b^5*x^3+35/2*a^4*b^4*x^4+56/5*a^5*b^3*x^5+14/3
*a^6*b^2*x^6+8/7*a^7*b*x^7+1/8*a^8*x^8+b^8*ln(x)

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Maxima [A]  time = 1.42842, size = 116, normalized size = 1.18 \[ \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^7,x, algorithm="maxima")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4
*x^4 + 56/3*a^3*b^5*x^3 + 14*a^2*b^6*x^2 + 8*a*b^7*x + b^8*log(x)

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Fricas [A]  time = 0.219125, size = 116, normalized size = 1.18 \[ \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^7,x, algorithm="fricas")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4
*x^4 + 56/3*a^3*b^5*x^3 + 14*a^2*b^6*x^2 + 8*a*b^7*x + b^8*log(x)

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Sympy [A]  time = 1.36282, size = 100, normalized size = 1.02 \[ \frac{a^{8} x^{8}}{8} + \frac{8 a^{7} b x^{7}}{7} + \frac{14 a^{6} b^{2} x^{6}}{3} + \frac{56 a^{5} b^{3} x^{5}}{5} + \frac{35 a^{4} b^{4} x^{4}}{2} + \frac{56 a^{3} b^{5} x^{3}}{3} + 14 a^{2} b^{6} x^{2} + 8 a b^{7} x + b^{8} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**7,x)

[Out]

a**8*x**8/8 + 8*a**7*b*x**7/7 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**5/5 + 35*a
**4*b**4*x**4/2 + 56*a**3*b**5*x**3/3 + 14*a**2*b**6*x**2 + 8*a*b**7*x + b**8*lo
g(x)

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GIAC/XCAS [A]  time = 0.228094, size = 117, normalized size = 1.19 \[ \frac{1}{8} \, a^{8} x^{8} + \frac{8}{7} \, a^{7} b x^{7} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{35}{2} \, a^{4} b^{4} x^{4} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 14 \, a^{2} b^{6} x^{2} + 8 \, a b^{7} x + b^{8}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^7,x, algorithm="giac")

[Out]

1/8*a^8*x^8 + 8/7*a^7*b*x^7 + 14/3*a^6*b^2*x^6 + 56/5*a^5*b^3*x^5 + 35/2*a^4*b^4
*x^4 + 56/3*a^3*b^5*x^3 + 14*a^2*b^6*x^2 + 8*a*b^7*x + b^8*ln(abs(x))