3.617 \(\int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{(x+1)^{12}}{17 x^{17}}+\frac{5 (x+1)^{12}}{272 x^{16}}-\frac{(x+1)^{12}}{204 x^{15}}+\frac{(x+1)^{12}}{952 x^{14}}-\frac{(x+1)^{12}}{6188 x^{13}}+\frac{(x+1)^{12}}{74256 x^{12}} \]

[Out]

-(1 + x)^12/(17*x^17) + (5*(1 + x)^12)/(272*x^16) - (1 + x)^12/(204*x^15) + (1 +
x)^12/(952*x^14) - (1 + x)^12/(6188*x^13) + (1 + x)^12/(74256*x^12)

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Rubi [A]  time = 0.0507311, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{(x+1)^{12}}{17 x^{17}}+\frac{5 (x+1)^{12}}{272 x^{16}}-\frac{(x+1)^{12}}{204 x^{15}}+\frac{(x+1)^{12}}{952 x^{14}}-\frac{(x+1)^{12}}{6188 x^{13}}+\frac{(x+1)^{12}}{74256 x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-(1 + x)^12/(17*x^17) + (5*(1 + x)^12)/(272*x^16) - (1 + x)^12/(204*x^15) + (1 +
x)^12/(952*x^14) - (1 + x)^12/(6188*x^13) + (1 + x)^12/(74256*x^12)

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Rubi in Sympy [A]  time = 10.2645, size = 61, normalized size = 0.84 \[ \frac{\left (x + 1\right )^{12}}{74256 x^{12}} - \frac{\left (x + 1\right )^{12}}{6188 x^{13}} + \frac{\left (x + 1\right )^{12}}{952 x^{14}} - \frac{\left (x + 1\right )^{12}}{204 x^{15}} + \frac{5 \left (x + 1\right )^{12}}{272 x^{16}} - \frac{\left (x + 1\right )^{12}}{17 x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2+2*x+1)**5/x**18,x)

[Out]

(x + 1)**12/(74256*x**12) - (x + 1)**12/(6188*x**13) + (x + 1)**12/(952*x**14) -
(x + 1)**12/(204*x**15) + 5*(x + 1)**12/(272*x**16) - (x + 1)**12/(17*x**17)

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Mathematica [A]  time = 0.00422186, size = 81, normalized size = 1.11 \[ -\frac{1}{17 x^{17}}-\frac{11}{16 x^{16}}-\frac{11}{3 x^{15}}-\frac{165}{14 x^{14}}-\frac{330}{13 x^{13}}-\frac{77}{2 x^{12}}-\frac{42}{x^{11}}-\frac{33}{x^{10}}-\frac{55}{3 x^9}-\frac{55}{8 x^8}-\frac{11}{7 x^7}-\frac{1}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-1/(17*x^17) - 11/(16*x^16) - 11/(3*x^15) - 165/(14*x^14) - 330/(13*x^13) - 77/(
2*x^12) - 42/x^11 - 33/x^10 - 55/(3*x^9) - 55/(8*x^8) - 11/(7*x^7) - 1/(6*x^6)

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Maple [A]  time = 0.01, size = 62, normalized size = 0.9 \[ -{\frac{77}{2\,{x}^{12}}}-{\frac{11}{3\,{x}^{15}}}-{\frac{11}{16\,{x}^{16}}}-{\frac{330}{13\,{x}^{13}}}-{\frac{1}{6\,{x}^{6}}}-33\,{x}^{-10}-{\frac{1}{17\,{x}^{17}}}-{\frac{55}{8\,{x}^{8}}}-42\,{x}^{-11}-{\frac{165}{14\,{x}^{14}}}-{\frac{55}{3\,{x}^{9}}}-{\frac{11}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2+2*x+1)^5/x^18,x)

[Out]

-77/2/x^12-11/3/x^15-11/16/x^16-330/13/x^13-1/6/x^6-33/x^10-1/17/x^17-55/8/x^8-4
2/x^11-165/14/x^14-55/3/x^9-11/7/x^7

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Maxima [A]  time = 0.691497, size = 81, normalized size = 1.11 \[ -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^18,x, algorithm="maxima")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 31
18752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368
)/x^17

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Fricas [A]  time = 0.276059, size = 81, normalized size = 1.11 \[ -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^18,x, algorithm="fricas")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 31
18752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368
)/x^17

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Sympy [A]  time = 0.659637, size = 61, normalized size = 0.84 \[ - \frac{12376 x^{11} + 116688 x^{10} + 510510 x^{9} + 1361360 x^{8} + 2450448 x^{7} + 3118752 x^{6} + 2858856 x^{5} + 1884960 x^{4} + 875160 x^{3} + 272272 x^{2} + 51051 x + 4368}{74256 x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2+2*x+1)**5/x**18,x)

[Out]

-(12376*x**11 + 116688*x**10 + 510510*x**9 + 1361360*x**8 + 2450448*x**7 + 31187
52*x**6 + 2858856*x**5 + 1884960*x**4 + 875160*x**3 + 272272*x**2 + 51051*x + 43
68)/(74256*x**17)

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GIAC/XCAS [A]  time = 0.267583, size = 81, normalized size = 1.11 \[ -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2*x + 1)^5*(x + 1)/x^18,x, algorithm="giac")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 31
18752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368
)/x^17