3.1595 \(\int \left (a+\frac{b}{x}\right )^8 x \, dx\)

Optimal. Leaf size=95 \[ \frac{a^8 x^2}{2}+8 a^7 b x+28 a^6 b^2 \log (x)-\frac{56 a^5 b^3}{x}-\frac{35 a^4 b^4}{x^2}-\frac{56 a^3 b^5}{3 x^3}-\frac{7 a^2 b^6}{x^4}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{6 x^6} \]

[Out]

-b^8/(6*x^6) - (8*a*b^7)/(5*x^5) - (7*a^2*b^6)/x^4 - (56*a^3*b^5)/(3*x^3) - (35*
a^4*b^4)/x^2 - (56*a^5*b^3)/x + 8*a^7*b*x + (a^8*x^2)/2 + 28*a^6*b^2*Log[x]

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Rubi [A]  time = 0.113331, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^8 x^2}{2}+8 a^7 b x+28 a^6 b^2 \log (x)-\frac{56 a^5 b^3}{x}-\frac{35 a^4 b^4}{x^2}-\frac{56 a^3 b^5}{3 x^3}-\frac{7 a^2 b^6}{x^4}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x,x]

[Out]

-b^8/(6*x^6) - (8*a*b^7)/(5*x^5) - (7*a^2*b^6)/x^4 - (56*a^3*b^5)/(3*x^3) - (35*
a^4*b^4)/x^2 - (56*a^5*b^3)/x + 8*a^7*b*x + (a^8*x^2)/2 + 28*a^6*b^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x,x)

[Out]

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

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Mathematica [A]  time = 0.00892433, size = 95, normalized size = 1. \[ \frac{a^8 x^2}{2}+8 a^7 b x+28 a^6 b^2 \log (x)-\frac{56 a^5 b^3}{x}-\frac{35 a^4 b^4}{x^2}-\frac{56 a^3 b^5}{3 x^3}-\frac{7 a^2 b^6}{x^4}-\frac{8 a b^7}{5 x^5}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x,x]

[Out]

-b^8/(6*x^6) - (8*a*b^7)/(5*x^5) - (7*a^2*b^6)/x^4 - (56*a^3*b^5)/(3*x^3) - (35*
a^4*b^4)/x^2 - (56*a^5*b^3)/x + 8*a^7*b*x + (a^8*x^2)/2 + 28*a^6*b^2*Log[x]

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Maple [A]  time = 0.01, size = 88, normalized size = 0.9 \[ -{\frac{{b}^{8}}{6\,{x}^{6}}}-{\frac{8\,a{b}^{7}}{5\,{x}^{5}}}-7\,{\frac{{a}^{2}{b}^{6}}{{x}^{4}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,{x}^{3}}}-35\,{\frac{{a}^{4}{b}^{4}}{{x}^{2}}}-56\,{\frac{{a}^{5}{b}^{3}}{x}}+8\,{a}^{7}bx+{\frac{{a}^{8}{x}^{2}}{2}}+28\,{a}^{6}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x,x)

[Out]

-1/6*b^8/x^6-8/5*a*b^7/x^5-7*a^2*b^6/x^4-56/3*a^3*b^5/x^3-35*a^4*b^4/x^2-56*a^5*
b^3/x+8*a^7*b*x+1/2*a^8*x^2+28*a^6*b^2*ln(x)

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Maxima [A]  time = 1.44099, size = 119, normalized size = 1.25 \[ \frac{1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2} \log \left (x\right ) - \frac{1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*log(x) - 1/30*(1680*a^5*b^3*x^5 + 1050*a^4*
b^4*x^4 + 560*a^3*b^5*x^3 + 210*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6

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Fricas [A]  time = 0.221656, size = 124, normalized size = 1.31 \[ \frac{15 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 840 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 1680 \, a^{5} b^{3} x^{5} - 1050 \, a^{4} b^{4} x^{4} - 560 \, a^{3} b^{5} x^{3} - 210 \, a^{2} b^{6} x^{2} - 48 \, a b^{7} x - 5 \, b^{8}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(15*a^8*x^8 + 240*a^7*b*x^7 + 840*a^6*b^2*x^6*log(x) - 1680*a^5*b^3*x^5 - 1
050*a^4*b^4*x^4 - 560*a^3*b^5*x^3 - 210*a^2*b^6*x^2 - 48*a*b^7*x - 5*b^8)/x^6

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Sympy [A]  time = 2.26207, size = 94, normalized size = 0.99 \[ \frac{a^{8} x^{2}}{2} + 8 a^{7} b x + 28 a^{6} b^{2} \log{\left (x \right )} - \frac{1680 a^{5} b^{3} x^{5} + 1050 a^{4} b^{4} x^{4} + 560 a^{3} b^{5} x^{3} + 210 a^{2} b^{6} x^{2} + 48 a b^{7} x + 5 b^{8}}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x,x)

[Out]

a**8*x**2/2 + 8*a**7*b*x + 28*a**6*b**2*log(x) - (1680*a**5*b**3*x**5 + 1050*a**
4*b**4*x**4 + 560*a**3*b**5*x**3 + 210*a**2*b**6*x**2 + 48*a*b**7*x + 5*b**8)/(3
0*x**6)

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GIAC/XCAS [A]  time = 0.225792, size = 120, normalized size = 1.26 \[ \frac{1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*ln(abs(x)) - 1/30*(1680*a^5*b^3*x^5 + 1050*
a^4*b^4*x^4 + 560*a^3*b^5*x^3 + 210*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6