3.1590 \(\int \left (a+\frac{b}{x}\right )^8 x^6 \, dx\)

Optimal. Leaf size=95 \[ \frac{a^8 x^7}{7}+\frac{4}{3} a^7 b x^6+\frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+8 a b^7 \log (x)-\frac{b^8}{x} \]

[Out]

-(b^8/x) + 28*a^2*b^6*x + 28*a^3*b^5*x^2 + (70*a^4*b^4*x^3)/3 + 14*a^5*b^3*x^4 +
(28*a^6*b^2*x^5)/5 + (4*a^7*b*x^6)/3 + (a^8*x^7)/7 + 8*a*b^7*Log[x]

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Rubi [A]  time = 0.106537, antiderivative size = 95, 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 \[ \frac{a^8 x^7}{7}+\frac{4}{3} a^7 b x^6+\frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+8 a b^7 \log (x)-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^6,x]

[Out]

-(b^8/x) + 28*a^2*b^6*x + 28*a^3*b^5*x^2 + (70*a^4*b^4*x^3)/3 + 14*a^5*b^3*x^4 +
(28*a^6*b^2*x^5)/5 + (4*a^7*b*x^6)/3 + (a^8*x^7)/7 + 8*a*b^7*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} x^{7}}{7} + \frac{4 a^{7} b x^{6}}{3} + \frac{28 a^{6} b^{2} x^{5}}{5} + 14 a^{5} b^{3} x^{4} + \frac{70 a^{4} b^{4} x^{3}}{3} + 56 a^{3} b^{5} \int x\, dx + 28 a^{2} b^{6} x + 8 a b^{7} \log{\left (x \right )} - \frac{b^{8}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**6,x)

[Out]

a**8*x**7/7 + 4*a**7*b*x**6/3 + 28*a**6*b**2*x**5/5 + 14*a**5*b**3*x**4 + 70*a**
4*b**4*x**3/3 + 56*a**3*b**5*Integral(x, x) + 28*a**2*b**6*x + 8*a*b**7*log(x) -
b**8/x

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Mathematica [A]  time = 0.0155233, size = 95, normalized size = 1. \[ \frac{a^8 x^7}{7}+\frac{4}{3} a^7 b x^6+\frac{28}{5} a^6 b^2 x^5+14 a^5 b^3 x^4+\frac{70}{3} a^4 b^4 x^3+28 a^3 b^5 x^2+28 a^2 b^6 x+8 a b^7 \log (x)-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^6,x]

[Out]

-(b^8/x) + 28*a^2*b^6*x + 28*a^3*b^5*x^2 + (70*a^4*b^4*x^3)/3 + 14*a^5*b^3*x^4 +
(28*a^6*b^2*x^5)/5 + (4*a^7*b*x^6)/3 + (a^8*x^7)/7 + 8*a*b^7*Log[x]

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Maple [A]  time = 0.009, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{8}}{x}}+28\,{a}^{2}{b}^{6}x+28\,{a}^{3}{b}^{5}{x}^{2}+{\frac{70\,{a}^{4}{b}^{4}{x}^{3}}{3}}+14\,{a}^{5}{b}^{3}{x}^{4}+{\frac{28\,{a}^{6}{b}^{2}{x}^{5}}{5}}+{\frac{4\,{a}^{7}b{x}^{6}}{3}}+{\frac{{a}^{8}{x}^{7}}{7}}+8\,a{b}^{7}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^6,x)

[Out]

-b^8/x+28*a^2*b^6*x+28*a^3*b^5*x^2+70/3*a^4*b^4*x^3+14*a^5*b^3*x^4+28/5*a^6*b^2*
x^5+4/3*a^7*b*x^6+1/7*a^8*x^7+8*a*b^7*ln(x)

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Maxima [A]  time = 1.43913, size = 117, normalized size = 1.23 \[ \frac{1}{7} \, a^{8} x^{7} + \frac{4}{3} \, a^{7} b x^{6} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 14 \, a^{5} b^{3} x^{4} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 28 \, a^{3} b^{5} x^{2} + 28 \, a^{2} b^{6} x + 8 \, a b^{7} \log \left (x\right ) - \frac{b^{8}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^6,x, algorithm="maxima")

[Out]

1/7*a^8*x^7 + 4/3*a^7*b*x^6 + 28/5*a^6*b^2*x^5 + 14*a^5*b^3*x^4 + 70/3*a^4*b^4*x
^3 + 28*a^3*b^5*x^2 + 28*a^2*b^6*x + 8*a*b^7*log(x) - b^8/x

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Fricas [A]  time = 0.219128, size = 124, normalized size = 1.31 \[ \frac{15 \, a^{8} x^{8} + 140 \, a^{7} b x^{7} + 588 \, a^{6} b^{2} x^{6} + 1470 \, a^{5} b^{3} x^{5} + 2450 \, a^{4} b^{4} x^{4} + 2940 \, a^{3} b^{5} x^{3} + 2940 \, a^{2} b^{6} x^{2} + 840 \, a b^{7} x \log \left (x\right ) - 105 \, b^{8}}{105 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^6,x, algorithm="fricas")

[Out]

1/105*(15*a^8*x^8 + 140*a^7*b*x^7 + 588*a^6*b^2*x^6 + 1470*a^5*b^3*x^5 + 2450*a^
4*b^4*x^4 + 2940*a^3*b^5*x^3 + 2940*a^2*b^6*x^2 + 840*a*b^7*x*log(x) - 105*b^8)/
x

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Sympy [A]  time = 1.40335, size = 95, normalized size = 1. \[ \frac{a^{8} x^{7}}{7} + \frac{4 a^{7} b x^{6}}{3} + \frac{28 a^{6} b^{2} x^{5}}{5} + 14 a^{5} b^{3} x^{4} + \frac{70 a^{4} b^{4} x^{3}}{3} + 28 a^{3} b^{5} x^{2} + 28 a^{2} b^{6} x + 8 a b^{7} \log{\left (x \right )} - \frac{b^{8}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**6,x)

[Out]

a**8*x**7/7 + 4*a**7*b*x**6/3 + 28*a**6*b**2*x**5/5 + 14*a**5*b**3*x**4 + 70*a**
4*b**4*x**3/3 + 28*a**3*b**5*x**2 + 28*a**2*b**6*x + 8*a*b**7*log(x) - b**8/x

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GIAC/XCAS [A]  time = 0.227876, size = 119, normalized size = 1.25 \[ \frac{1}{7} \, a^{8} x^{7} + \frac{4}{3} \, a^{7} b x^{6} + \frac{28}{5} \, a^{6} b^{2} x^{5} + 14 \, a^{5} b^{3} x^{4} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 28 \, a^{3} b^{5} x^{2} + 28 \, a^{2} b^{6} x + 8 \, a b^{7}{\rm ln}\left ({\left | x \right |}\right ) - \frac{b^{8}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^6,x, algorithm="giac")

[Out]

1/7*a^8*x^7 + 4/3*a^7*b*x^6 + 28/5*a^6*b^2*x^5 + 14*a^5*b^3*x^4 + 70/3*a^4*b^4*x
^3 + 28*a^3*b^5*x^2 + 28*a^2*b^6*x + 8*a*b^7*ln(abs(x)) - b^8/x