3.607 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^8} \, dx\)

Optimal. Leaf size=70 \[ -\frac{1}{7 x^7}-\frac{11}{6 x^6}-\frac{11}{x^5}+\frac{x^4}{4}-\frac{165}{4 x^4}+\frac{11 x^3}{3}-\frac{110}{x^3}+\frac{55 x^2}{2}-\frac{231}{x^2}+165 x-\frac{462}{x}+330 \log (x) \]

[Out]

-1/(7*x^7) - 11/(6*x^6) - 11/x^5 - 165/(4*x^4) - 110/x^3 - 231/x^2 - 462/x + 165
*x + (55*x^2)/2 + (11*x^3)/3 + x^4/4 + 330*Log[x]

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Rubi [A]  time = 0.0510814, 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 \[ -\frac{1}{7 x^7}-\frac{11}{6 x^6}-\frac{11}{x^5}+\frac{x^4}{4}-\frac{165}{4 x^4}+\frac{11 x^3}{3}-\frac{110}{x^3}+\frac{55 x^2}{2}-\frac{231}{x^2}+165 x-\frac{462}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-1/(7*x^7) - 11/(6*x^6) - 11/x^5 - 165/(4*x^4) - 110/x^3 - 231/x^2 - 462/x + 165
*x + (55*x^2)/2 + (11*x^3)/3 + x^4/4 + 330*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{4}}{4} + \frac{11 x^{3}}{3} + 165 x + 330 \log{\left (x \right )} + 55 \int x\, dx - \frac{462}{x} - \frac{231}{x^{2}} - \frac{110}{x^{3}} - \frac{165}{4 x^{4}} - \frac{11}{x^{5}} - \frac{11}{6 x^{6}} - \frac{1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**8,x)

[Out]

x**4/4 + 11*x**3/3 + 165*x + 330*log(x) + 55*Integral(x, x) - 462/x - 231/x**2 -
110/x**3 - 165/(4*x**4) - 11/x**5 - 11/(6*x**6) - 1/(7*x**7)

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Mathematica [A]  time = 0.00447336, size = 70, normalized size = 1. \[ -\frac{1}{7 x^7}-\frac{11}{6 x^6}-\frac{11}{x^5}+\frac{x^4}{4}-\frac{165}{4 x^4}+\frac{11 x^3}{3}-\frac{110}{x^3}+\frac{55 x^2}{2}-\frac{231}{x^2}+165 x-\frac{462}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-1/(7*x^7) - 11/(6*x^6) - 11/x^5 - 165/(4*x^4) - 110/x^3 - 231/x^2 - 462/x + 165
*x + (55*x^2)/2 + (11*x^3)/3 + x^4/4 + 330*Log[x]

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Maple [A]  time = 0.009, size = 59, normalized size = 0.8 \[ -{\frac{1}{7\,{x}^{7}}}-{\frac{11}{6\,{x}^{6}}}-11\,{x}^{-5}-{\frac{165}{4\,{x}^{4}}}-110\,{x}^{-3}-231\,{x}^{-2}-462\,{x}^{-1}+165\,x+{\frac{55\,{x}^{2}}{2}}+{\frac{11\,{x}^{3}}{3}}+{\frac{{x}^{4}}{4}}+330\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^8,x)

[Out]

-1/7/x^7-11/6/x^6-11/x^5-165/4/x^4-110/x^3-231/x^2-462/x+165*x+55/2*x^2+11/3*x^3
+1/4*x^4+330*ln(x)

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Maxima [A]  time = 0.687505, size = 78, normalized size = 1.11 \[ \frac{1}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{55}{2} \, x^{2} + 165 \, x - \frac{38808 \, x^{6} + 19404 \, x^{5} + 9240 \, x^{4} + 3465 \, x^{3} + 924 \, x^{2} + 154 \, x + 12}{84 \, x^{7}} + 330 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^8,x, algorithm="maxima")

[Out]

1/4*x^4 + 11/3*x^3 + 55/2*x^2 + 165*x - 1/84*(38808*x^6 + 19404*x^5 + 9240*x^4 +
3465*x^3 + 924*x^2 + 154*x + 12)/x^7 + 330*log(x)

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Fricas [A]  time = 0.277967, size = 84, normalized size = 1.2 \[ \frac{21 \, x^{11} + 308 \, x^{10} + 2310 \, x^{9} + 13860 \, x^{8} + 27720 \, x^{7} \log \left (x\right ) - 38808 \, x^{6} - 19404 \, x^{5} - 9240 \, x^{4} - 3465 \, x^{3} - 924 \, x^{2} - 154 \, x - 12}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^8,x, algorithm="fricas")

[Out]

1/84*(21*x^11 + 308*x^10 + 2310*x^9 + 13860*x^8 + 27720*x^7*log(x) - 38808*x^6 -
19404*x^5 - 9240*x^4 - 3465*x^3 - 924*x^2 - 154*x - 12)/x^7

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Sympy [A]  time = 0.369708, size = 61, normalized size = 0.87 \[ \frac{x^{4}}{4} + \frac{11 x^{3}}{3} + \frac{55 x^{2}}{2} + 165 x + 330 \log{\left (x \right )} - \frac{38808 x^{6} + 19404 x^{5} + 9240 x^{4} + 3465 x^{3} + 924 x^{2} + 154 x + 12}{84 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**8,x)

[Out]

x**4/4 + 11*x**3/3 + 55*x**2/2 + 165*x + 330*log(x) - (38808*x**6 + 19404*x**5 +
9240*x**4 + 3465*x**3 + 924*x**2 + 154*x + 12)/(84*x**7)

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GIAC/XCAS [A]  time = 0.268399, size = 80, normalized size = 1.14 \[ \frac{1}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{55}{2} \, x^{2} + 165 \, x - \frac{38808 \, x^{6} + 19404 \, x^{5} + 9240 \, x^{4} + 3465 \, x^{3} + 924 \, x^{2} + 154 \, x + 12}{84 \, x^{7}} + 330 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^8,x, algorithm="giac")

[Out]

1/4*x^4 + 11/3*x^3 + 55/2*x^2 + 165*x - 1/84*(38808*x^6 + 19404*x^5 + 9240*x^4 +
3465*x^3 + 924*x^2 + 154*x + 12)/x^7 + 330*ln(abs(x))