3.606 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0495558, 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 \[ -\frac{1}{6 x^6}+\frac{x^5}{5}-\frac{11}{5 x^5}+\frac{11 x^4}{4}-\frac{55}{4 x^4}+\frac{55 x^3}{3}-\frac{55}{x^3}+\frac{165 x^2}{2}-\frac{165}{x^2}+330 x-\frac{462}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00433225, size = 72, normalized size = 1. \[ -\frac{1}{6 x^6}+\frac{x^5}{5}-\frac{11}{5 x^5}+\frac{11 x^4}{4}-\frac{55}{4 x^4}+\frac{55 x^3}{3}-\frac{55}{x^3}+\frac{165 x^2}{2}-\frac{165}{x^2}+330 x-\frac{462}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^7,x)

[Out]

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

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Maxima [A]  time = 0.696413, size = 78, normalized size = 1.08 \[ \frac{1}{5} \, x^{5} + \frac{11}{4} \, x^{4} + \frac{55}{3} \, x^{3} + \frac{165}{2} \, x^{2} + 330 \, x - \frac{27720 \, x^{5} + 9900 \, x^{4} + 3300 \, x^{3} + 825 \, x^{2} + 132 \, x + 10}{60 \, x^{6}} + 462 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^7,x, algorithm="maxima")

[Out]

1/5*x^5 + 11/4*x^4 + 55/3*x^3 + 165/2*x^2 + 330*x - 1/60*(27720*x^5 + 9900*x^4 +
3300*x^3 + 825*x^2 + 132*x + 10)/x^6 + 462*log(x)

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Fricas [A]  time = 0.298408, size = 84, normalized size = 1.17 \[ \frac{12 \, x^{11} + 165 \, x^{10} + 1100 \, x^{9} + 4950 \, x^{8} + 19800 \, x^{7} + 27720 \, x^{6} \log \left (x\right ) - 27720 \, x^{5} - 9900 \, x^{4} - 3300 \, x^{3} - 825 \, x^{2} - 132 \, x - 10}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.343629, size = 63, normalized size = 0.88 \[ \frac{x^{5}}{5} + \frac{11 x^{4}}{4} + \frac{55 x^{3}}{3} + \frac{165 x^{2}}{2} + 330 x + 462 \log{\left (x \right )} - \frac{27720 x^{5} + 9900 x^{4} + 3300 x^{3} + 825 x^{2} + 132 x + 10}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**7,x)

[Out]

x**5/5 + 11*x**4/4 + 55*x**3/3 + 165*x**2/2 + 330*x + 462*log(x) - (27720*x**5 +
9900*x**4 + 3300*x**3 + 825*x**2 + 132*x + 10)/(60*x**6)

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GIAC/XCAS [A]  time = 0.272736, size = 80, normalized size = 1.11 \[ \frac{1}{5} \, x^{5} + \frac{11}{4} \, x^{4} + \frac{55}{3} \, x^{3} + \frac{165}{2} \, x^{2} + 330 \, x - \frac{27720 \, x^{5} + 9900 \, x^{4} + 3300 \, x^{3} + 825 \, x^{2} + 132 \, x + 10}{60 \, x^{6}} + 462 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^7,x, algorithm="giac")

[Out]

1/5*x^5 + 11/4*x^4 + 55/3*x^3 + 165/2*x^2 + 330*x - 1/60*(27720*x^5 + 9900*x^4 +
3300*x^3 + 825*x^2 + 132*x + 10)/x^6 + 462*ln(abs(x))