3.608 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^9} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{8 x^8}-\frac{11}{7 x^7}-\frac{55}{6 x^6}-\frac{33}{x^5}-\frac{165}{2 x^4}+\frac{x^3}{3}-\frac{154}{x^3}+\frac{11 x^2}{2}-\frac{231}{x^2}+55 x-\frac{330}{x}+165 \log (x) \]

[Out]

-1/(8*x^8) - 11/(7*x^7) - 55/(6*x^6) - 33/x^5 - 165/(2*x^4) - 154/x^3 - 231/x^2
- 330/x + 55*x + (11*x^2)/2 + x^3/3 + 165*Log[x]

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Rubi [A]  time = 0.0494854, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{1}{8 x^8}-\frac{11}{7 x^7}-\frac{55}{6 x^6}-\frac{33}{x^5}-\frac{165}{2 x^4}+\frac{x^3}{3}-\frac{154}{x^3}+\frac{11 x^2}{2}-\frac{231}{x^2}+55 x-\frac{330}{x}+165 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-1/(8*x^8) - 11/(7*x^7) - 55/(6*x^6) - 33/x^5 - 165/(2*x^4) - 154/x^3 - 231/x^2
- 330/x + 55*x + (11*x^2)/2 + x^3/3 + 165*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3} + 55 x + 165 \log{\left (x \right )} + 11 \int x\, dx - \frac{330}{x} - \frac{231}{x^{2}} - \frac{154}{x^{3}} - \frac{165}{2 x^{4}} - \frac{33}{x^{5}} - \frac{55}{6 x^{6}} - \frac{11}{7 x^{7}} - \frac{1}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**9,x)

[Out]

x**3/3 + 55*x + 165*log(x) + 11*Integral(x, x) - 330/x - 231/x**2 - 154/x**3 - 1
65/(2*x**4) - 33/x**5 - 55/(6*x**6) - 11/(7*x**7) - 1/(8*x**8)

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Mathematica [A]  time = 0.00436649, size = 70, normalized size = 1. \[ -\frac{1}{8 x^8}-\frac{11}{7 x^7}-\frac{55}{6 x^6}-\frac{33}{x^5}-\frac{165}{2 x^4}+\frac{x^3}{3}-\frac{154}{x^3}+\frac{11 x^2}{2}-\frac{231}{x^2}+55 x-\frac{330}{x}+165 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-1/(8*x^8) - 11/(7*x^7) - 55/(6*x^6) - 33/x^5 - 165/(2*x^4) - 154/x^3 - 231/x^2
- 330/x + 55*x + (11*x^2)/2 + x^3/3 + 165*Log[x]

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Maple [A]  time = 0.01, size = 59, normalized size = 0.8 \[ -{\frac{1}{8\,{x}^{8}}}-{\frac{11}{7\,{x}^{7}}}-{\frac{55}{6\,{x}^{6}}}-33\,{x}^{-5}-{\frac{165}{2\,{x}^{4}}}-154\,{x}^{-3}-231\,{x}^{-2}-330\,{x}^{-1}+55\,x+{\frac{11\,{x}^{2}}{2}}+{\frac{{x}^{3}}{3}}+165\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^9,x)

[Out]

-1/8/x^8-11/7/x^7-55/6/x^6-33/x^5-165/2/x^4-154/x^3-231/x^2-330/x+55*x+11/2*x^2+
1/3*x^3+165*ln(x)

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Maxima [A]  time = 0.683337, size = 78, normalized size = 1.11 \[ \frac{1}{3} \, x^{3} + \frac{11}{2} \, x^{2} + 55 \, x - \frac{55440 \, x^{7} + 38808 \, x^{6} + 25872 \, x^{5} + 13860 \, x^{4} + 5544 \, x^{3} + 1540 \, x^{2} + 264 \, x + 21}{168 \, x^{8}} + 165 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^9,x, algorithm="maxima")

[Out]

1/3*x^3 + 11/2*x^2 + 55*x - 1/168*(55440*x^7 + 38808*x^6 + 25872*x^5 + 13860*x^4
+ 5544*x^3 + 1540*x^2 + 264*x + 21)/x^8 + 165*log(x)

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Fricas [A]  time = 0.310024, size = 84, normalized size = 1.2 \[ \frac{56 \, x^{11} + 924 \, x^{10} + 9240 \, x^{9} + 27720 \, x^{8} \log \left (x\right ) - 55440 \, x^{7} - 38808 \, x^{6} - 25872 \, x^{5} - 13860 \, x^{4} - 5544 \, x^{3} - 1540 \, x^{2} - 264 \, x - 21}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^9,x, algorithm="fricas")

[Out]

1/168*(56*x^11 + 924*x^10 + 9240*x^9 + 27720*x^8*log(x) - 55440*x^7 - 38808*x^6
- 25872*x^5 - 13860*x^4 - 5544*x^3 - 1540*x^2 - 264*x - 21)/x^8

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Sympy [A]  time = 0.41901, size = 60, normalized size = 0.86 \[ \frac{x^{3}}{3} + \frac{11 x^{2}}{2} + 55 x + 165 \log{\left (x \right )} - \frac{55440 x^{7} + 38808 x^{6} + 25872 x^{5} + 13860 x^{4} + 5544 x^{3} + 1540 x^{2} + 264 x + 21}{168 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**9,x)

[Out]

x**3/3 + 11*x**2/2 + 55*x + 165*log(x) - (55440*x**7 + 38808*x**6 + 25872*x**5 +
13860*x**4 + 5544*x**3 + 1540*x**2 + 264*x + 21)/(168*x**8)

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GIAC/XCAS [A]  time = 0.27246, size = 80, normalized size = 1.14 \[ \frac{1}{3} \, x^{3} + \frac{11}{2} \, x^{2} + 55 \, x - \frac{55440 \, x^{7} + 38808 \, x^{6} + 25872 \, x^{5} + 13860 \, x^{4} + 5544 \, x^{3} + 1540 \, x^{2} + 264 \, x + 21}{168 \, x^{8}} + 165 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^9,x, algorithm="giac")

[Out]

1/3*x^3 + 11/2*x^2 + 55*x - 1/168*(55440*x^7 + 38808*x^6 + 25872*x^5 + 13860*x^4
+ 5544*x^3 + 1540*x^2 + 264*x + 21)/x^8 + 165*ln(abs(x))