3.1594 \(\int \left (a+\frac{b}{x}\right )^8 x^2 \, dx\)

Optimal. Leaf size=93 \[ \frac{a^8 x^3}{3}+4 a^7 b x^2+28 a^6 b^2 x+56 a^5 b^3 \log (x)-\frac{70 a^4 b^4}{x}-\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

[Out]

-b^8/(5*x^5) - (2*a*b^7)/x^4 - (28*a^2*b^6)/(3*x^3) - (28*a^3*b^5)/x^2 - (70*a^4
*b^4)/x + 28*a^6*b^2*x + 4*a^7*b*x^2 + (a^8*x^3)/3 + 56*a^5*b^3*Log[x]

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Rubi [A]  time = 0.109083, antiderivative size = 93, 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 \[ \frac{a^8 x^3}{3}+4 a^7 b x^2+28 a^6 b^2 x+56 a^5 b^3 \log (x)-\frac{70 a^4 b^4}{x}-\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8*x^2,x]

[Out]

-b^8/(5*x^5) - (2*a*b^7)/x^4 - (28*a^2*b^6)/(3*x^3) - (28*a^3*b^5)/x^2 - (70*a^4
*b^4)/x + 28*a^6*b^2*x + 4*a^7*b*x^2 + (a^8*x^3)/3 + 56*a^5*b^3*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8*x**2,x)

[Out]

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

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Mathematica [A]  time = 0.015188, size = 93, normalized size = 1. \[ \frac{a^8 x^3}{3}+4 a^7 b x^2+28 a^6 b^2 x+56 a^5 b^3 \log (x)-\frac{70 a^4 b^4}{x}-\frac{28 a^3 b^5}{x^2}-\frac{28 a^2 b^6}{3 x^3}-\frac{2 a b^7}{x^4}-\frac{b^8}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8*x^2,x]

[Out]

-b^8/(5*x^5) - (2*a*b^7)/x^4 - (28*a^2*b^6)/(3*x^3) - (28*a^3*b^5)/x^2 - (70*a^4
*b^4)/x + 28*a^6*b^2*x + 4*a^7*b*x^2 + (a^8*x^3)/3 + 56*a^5*b^3*Log[x]

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Maple [A]  time = 0.01, size = 88, normalized size = 1. \[ -{\frac{{b}^{8}}{5\,{x}^{5}}}-2\,{\frac{a{b}^{7}}{{x}^{4}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}-28\,{\frac{{a}^{3}{b}^{5}}{{x}^{2}}}-70\,{\frac{{a}^{4}{b}^{4}}{x}}+28\,{a}^{6}{b}^{2}x+4\,{a}^{7}b{x}^{2}+{\frac{{a}^{8}{x}^{3}}{3}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8*x^2,x)

[Out]

-1/5*b^8/x^5-2*a*b^7/x^4-28/3*a^2*b^6/x^3-28*a^3*b^5/x^2-70*a^4*b^4/x+28*a^6*b^2
*x+4*a^7*b*x^2+1/3*a^8*x^3+56*a^5*b^3*ln(x)

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Maxima [A]  time = 1.42995, size = 119, normalized size = 1.28 \[ \frac{1}{3} \, a^{8} x^{3} + 4 \, a^{7} b x^{2} + 28 \, a^{6} b^{2} x + 56 \, a^{5} b^{3} \log \left (x\right ) - \frac{1050 \, a^{4} b^{4} x^{4} + 420 \, a^{3} b^{5} x^{3} + 140 \, a^{2} b^{6} x^{2} + 30 \, a b^{7} x + 3 \, b^{8}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^2,x, algorithm="maxima")

[Out]

1/3*a^8*x^3 + 4*a^7*b*x^2 + 28*a^6*b^2*x + 56*a^5*b^3*log(x) - 1/15*(1050*a^4*b^
4*x^4 + 420*a^3*b^5*x^3 + 140*a^2*b^6*x^2 + 30*a*b^7*x + 3*b^8)/x^5

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Fricas [A]  time = 0.220996, size = 124, normalized size = 1.33 \[ \frac{5 \, a^{8} x^{8} + 60 \, a^{7} b x^{7} + 420 \, a^{6} b^{2} x^{6} + 840 \, a^{5} b^{3} x^{5} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{4} - 420 \, a^{3} b^{5} x^{3} - 140 \, a^{2} b^{6} x^{2} - 30 \, a b^{7} x - 3 \, b^{8}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^2,x, algorithm="fricas")

[Out]

1/15*(5*a^8*x^8 + 60*a^7*b*x^7 + 420*a^6*b^2*x^6 + 840*a^5*b^3*x^5*log(x) - 1050
*a^4*b^4*x^4 - 420*a^3*b^5*x^3 - 140*a^2*b^6*x^2 - 30*a*b^7*x - 3*b^8)/x^5

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Sympy [A]  time = 2.43273, size = 94, normalized size = 1.01 \[ \frac{a^{8} x^{3}}{3} + 4 a^{7} b x^{2} + 28 a^{6} b^{2} x + 56 a^{5} b^{3} \log{\left (x \right )} - \frac{1050 a^{4} b^{4} x^{4} + 420 a^{3} b^{5} x^{3} + 140 a^{2} b^{6} x^{2} + 30 a b^{7} x + 3 b^{8}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8*x**2,x)

[Out]

a**8*x**3/3 + 4*a**7*b*x**2 + 28*a**6*b**2*x + 56*a**5*b**3*log(x) - (1050*a**4*
b**4*x**4 + 420*a**3*b**5*x**3 + 140*a**2*b**6*x**2 + 30*a*b**7*x + 3*b**8)/(15*
x**5)

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GIAC/XCAS [A]  time = 0.225254, size = 120, normalized size = 1.29 \[ \frac{1}{3} \, a^{8} x^{3} + 4 \, a^{7} b x^{2} + 28 \, a^{6} b^{2} x + 56 \, a^{5} b^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1050 \, a^{4} b^{4} x^{4} + 420 \, a^{3} b^{5} x^{3} + 140 \, a^{2} b^{6} x^{2} + 30 \, a b^{7} x + 3 \, b^{8}}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8*x^2,x, algorithm="giac")

[Out]

1/3*a^8*x^3 + 4*a^7*b*x^2 + 28*a^6*b^2*x + 56*a^5*b^3*ln(abs(x)) - 1/15*(1050*a^
4*b^4*x^4 + 420*a^3*b^5*x^3 + 140*a^2*b^6*x^2 + 30*a*b^7*x + 3*b^8)/x^5