3.605 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^6} \, dx\)

Optimal. Leaf size=72 \[ \frac{x^6}{6}+\frac{11 x^5}{5}-\frac{1}{5 x^5}+\frac{55 x^4}{4}-\frac{11}{4 x^4}+55 x^3-\frac{55}{3 x^3}+165 x^2-\frac{165}{2 x^2}+462 x-\frac{330}{x}+462 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0494163, antiderivative size = 72, 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 \[ \frac{x^6}{6}+\frac{11 x^5}{5}-\frac{1}{5 x^5}+\frac{55 x^4}{4}-\frac{11}{4 x^4}+55 x^3-\frac{55}{3 x^3}+165 x^2-\frac{165}{2 x^2}+462 x-\frac{330}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.00467399, size = 72, normalized size = 1. \[ \frac{x^6}{6}+\frac{11 x^5}{5}-\frac{1}{5 x^5}+\frac{55 x^4}{4}-\frac{11}{4 x^4}+55 x^3-\frac{55}{3 x^3}+165 x^2-\frac{165}{2 x^2}+462 x-\frac{330}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

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

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Maple [A]  time = 0.01, size = 59, normalized size = 0.8 \[ -{\frac{1}{5\,{x}^{5}}}-{\frac{11}{4\,{x}^{4}}}-{\frac{55}{3\,{x}^{3}}}-{\frac{165}{2\,{x}^{2}}}-330\,{x}^{-1}+462\,x+165\,{x}^{2}+55\,{x}^{3}+{\frac{55\,{x}^{4}}{4}}+{\frac{11\,{x}^{5}}{5}}+{\frac{{x}^{6}}{6}}+462\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^6,x)

[Out]

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

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Maxima [A]  time = 0.691008, size = 78, normalized size = 1.08 \[ \frac{1}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{55}{4} \, x^{4} + 55 \, x^{3} + 165 \, x^{2} + 462 \, x - \frac{19800 \, x^{4} + 4950 \, x^{3} + 1100 \, x^{2} + 165 \, x + 12}{60 \, x^{5}} + 462 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^6,x, algorithm="maxima")

[Out]

1/6*x^6 + 11/5*x^5 + 55/4*x^4 + 55*x^3 + 165*x^2 + 462*x - 1/60*(19800*x^4 + 495
0*x^3 + 1100*x^2 + 165*x + 12)/x^5 + 462*log(x)

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Fricas [A]  time = 0.303503, size = 84, normalized size = 1.17 \[ \frac{10 \, x^{11} + 132 \, x^{10} + 825 \, x^{9} + 3300 \, x^{8} + 9900 \, x^{7} + 27720 \, x^{6} + 27720 \, x^{5} \log \left (x\right ) - 19800 \, x^{4} - 4950 \, x^{3} - 1100 \, x^{2} - 165 \, x - 12}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^6,x, algorithm="fricas")

[Out]

1/60*(10*x^11 + 132*x^10 + 825*x^9 + 3300*x^8 + 9900*x^7 + 27720*x^6 + 27720*x^5
*log(x) - 19800*x^4 - 4950*x^3 - 1100*x^2 - 165*x - 12)/x^5

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Sympy [A]  time = 0.326388, size = 61, normalized size = 0.85 \[ \frac{x^{6}}{6} + \frac{11 x^{5}}{5} + \frac{55 x^{4}}{4} + 55 x^{3} + 165 x^{2} + 462 x + 462 \log{\left (x \right )} - \frac{19800 x^{4} + 4950 x^{3} + 1100 x^{2} + 165 x + 12}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**6,x)

[Out]

x**6/6 + 11*x**5/5 + 55*x**4/4 + 55*x**3 + 165*x**2 + 462*x + 462*log(x) - (1980
0*x**4 + 4950*x**3 + 1100*x**2 + 165*x + 12)/(60*x**5)

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GIAC/XCAS [A]  time = 0.27536, size = 80, normalized size = 1.11 \[ \frac{1}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{55}{4} \, x^{4} + 55 \, x^{3} + 165 \, x^{2} + 462 \, x - \frac{19800 \, x^{4} + 4950 \, x^{3} + 1100 \, x^{2} + 165 \, x + 12}{60 \, x^{5}} + 462 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^6,x, algorithm="giac")

[Out]

1/6*x^6 + 11/5*x^5 + 55/4*x^4 + 55*x^3 + 165*x^2 + 462*x - 1/60*(19800*x^4 + 495
0*x^3 + 1100*x^2 + 165*x + 12)/x^5 + 462*ln(abs(x))