3.620 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{1}{20 x^{20}}-\frac{11}{19 x^{19}}-\frac{55}{18 x^{18}}-\frac{165}{17 x^{17}}-\frac{165}{8 x^{16}}-\frac{154}{5 x^{15}}-\frac{33}{x^{14}}-\frac{330}{13 x^{13}}-\frac{55}{4 x^{12}}-\frac{5}{x^{11}}-\frac{11}{10 x^{10}}-\frac{1}{9 x^9} \]

[Out]

-1/(20*x^20) - 11/(19*x^19) - 55/(18*x^18) - 165/(17*x^17) - 165/(8*x^16) - 154/
(5*x^15) - 33/x^14 - 330/(13*x^13) - 55/(4*x^12) - 5/x^11 - 11/(10*x^10) - 1/(9*
x^9)

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Rubi [A]  time = 0.0526298, antiderivative size = 81, 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 \[ -\frac{1}{20 x^{20}}-\frac{11}{19 x^{19}}-\frac{55}{18 x^{18}}-\frac{165}{17 x^{17}}-\frac{165}{8 x^{16}}-\frac{154}{5 x^{15}}-\frac{33}{x^{14}}-\frac{330}{13 x^{13}}-\frac{55}{4 x^{12}}-\frac{5}{x^{11}}-\frac{11}{10 x^{10}}-\frac{1}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^21,x]

[Out]

-1/(20*x^20) - 11/(19*x^19) - 55/(18*x^18) - 165/(17*x^17) - 165/(8*x^16) - 154/
(5*x^15) - 33/x^14 - 330/(13*x^13) - 55/(4*x^12) - 5/x^11 - 11/(10*x^10) - 1/(9*
x^9)

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Rubi in Sympy [A]  time = 11.8761, size = 78, normalized size = 0.96 \[ - \frac{1}{9 x^{9}} - \frac{11}{10 x^{10}} - \frac{5}{x^{11}} - \frac{55}{4 x^{12}} - \frac{330}{13 x^{13}} - \frac{33}{x^{14}} - \frac{154}{5 x^{15}} - \frac{165}{8 x^{16}} - \frac{165}{17 x^{17}} - \frac{55}{18 x^{18}} - \frac{11}{19 x^{19}} - \frac{1}{20 x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**21,x)

[Out]

-1/(9*x**9) - 11/(10*x**10) - 5/x**11 - 55/(4*x**12) - 330/(13*x**13) - 33/x**14
- 154/(5*x**15) - 165/(8*x**16) - 165/(17*x**17) - 55/(18*x**18) - 11/(19*x**19
) - 1/(20*x**20)

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Mathematica [A]  time = 0.0041761, size = 81, normalized size = 1. \[ -\frac{1}{20 x^{20}}-\frac{11}{19 x^{19}}-\frac{55}{18 x^{18}}-\frac{165}{17 x^{17}}-\frac{165}{8 x^{16}}-\frac{154}{5 x^{15}}-\frac{33}{x^{14}}-\frac{330}{13 x^{13}}-\frac{55}{4 x^{12}}-\frac{5}{x^{11}}-\frac{11}{10 x^{10}}-\frac{1}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^21,x]

[Out]

-1/(20*x^20) - 11/(19*x^19) - 55/(18*x^18) - 165/(17*x^17) - 165/(8*x^16) - 154/
(5*x^15) - 33/x^14 - 330/(13*x^13) - 55/(4*x^12) - 5/x^11 - 11/(10*x^10) - 1/(9*
x^9)

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Maple [A]  time = 0.009, size = 62, normalized size = 0.8 \[ -{\frac{1}{20\,{x}^{20}}}-{\frac{11}{19\,{x}^{19}}}-{\frac{55}{18\,{x}^{18}}}-{\frac{165}{17\,{x}^{17}}}-{\frac{165}{8\,{x}^{16}}}-{\frac{154}{5\,{x}^{15}}}-33\,{x}^{-14}-{\frac{330}{13\,{x}^{13}}}-{\frac{55}{4\,{x}^{12}}}-5\,{x}^{-11}-{\frac{11}{10\,{x}^{10}}}-{\frac{1}{9\,{x}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^21,x)

[Out]

-1/20/x^20-11/19/x^19-55/18/x^18-165/17/x^17-165/8/x^16-154/5/x^15-33/x^14-330/1
3/x^13-55/4/x^12-5/x^11-11/10/x^10-1/9/x^9

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Maxima [A]  time = 0.685746, size = 81, normalized size = 1. \[ -\frac{167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^21,x, algorithm="maxima")

[Out]

-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 38372400*x
^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 4618900*x^2 + 8
75160*x + 75582)/x^20

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Fricas [A]  time = 0.269699, size = 81, normalized size = 1. \[ -\frac{167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^21,x, algorithm="fricas")

[Out]

-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 38372400*x
^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 4618900*x^2 + 8
75160*x + 75582)/x^20

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Sympy [A]  time = 0.748556, size = 61, normalized size = 0.75 \[ - \frac{167960 x^{11} + 1662804 x^{10} + 7558200 x^{9} + 20785050 x^{8} + 38372400 x^{7} + 49884120 x^{6} + 46558512 x^{5} + 31177575 x^{4} + 14671800 x^{3} + 4618900 x^{2} + 875160 x + 75582}{1511640 x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**21,x)

[Out]

-(167960*x**11 + 1662804*x**10 + 7558200*x**9 + 20785050*x**8 + 38372400*x**7 +
49884120*x**6 + 46558512*x**5 + 31177575*x**4 + 14671800*x**3 + 4618900*x**2 + 8
75160*x + 75582)/(1511640*x**20)

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GIAC/XCAS [A]  time = 0.267422, size = 81, normalized size = 1. \[ -\frac{167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^21,x, algorithm="giac")

[Out]

-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 38372400*x
^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 4618900*x^2 + 8
75160*x + 75582)/x^20