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

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

[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 = 0.021086, 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, Rules used = {27, 43} \[ -\frac{1}{9 x^9}-\frac{11}{10 x^{10}}-\frac{5}{x^{11}}-\frac{55}{4 x^{12}}-\frac{330}{13 x^{13}}-\frac{33}{x^{14}}-\frac{154}{5 x^{15}}-\frac{165}{8 x^{16}}-\frac{165}{17 x^{17}}-\frac{55}{18 x^{18}}-\frac{11}{19 x^{19}}-\frac{1}{20 x^{20}} \]

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)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx &=\int \frac{(1+x)^{11}}{x^{21}} \, dx\\ &=\int \left (\frac{1}{x^{21}}+\frac{11}{x^{20}}+\frac{55}{x^{19}}+\frac{165}{x^{18}}+\frac{330}{x^{17}}+\frac{462}{x^{16}}+\frac{462}{x^{15}}+\frac{330}{x^{14}}+\frac{165}{x^{13}}+\frac{55}{x^{12}}+\frac{11}{x^{11}}+\frac{1}{x^{10}}\right ) \, dx\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.0022271, size = 81, normalized size = 1. \[ -\frac{1}{9 x^9}-\frac{11}{10 x^{10}}-\frac{5}{x^{11}}-\frac{55}{4 x^{12}}-\frac{330}{13 x^{13}}-\frac{33}{x^{14}}-\frac{154}{5 x^{15}}-\frac{165}{8 x^{16}}-\frac{165}{17 x^{17}}-\frac{55}{18 x^{18}}-\frac{11}{19 x^{19}}-\frac{1}{20 x^{20}} \]

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 = 0.007, size = 62, normalized size = 0.8 \begin{align*} -{\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}}} \end{align*}

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^14-330/13/x^13-55/4/x^12-5/x^11-11/10/
x^10-1/9/x^9

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Maxima [A]  time = 1.0051, size = 81, normalized size = 1. \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/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 + 875160*x + 75582)/x^20

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Fricas [A]  time = 1.24241, size = 247, normalized size = 3.05 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/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 + 875160*x + 75582)/x^20

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Sympy [A]  time = 0.227974, size = 61, normalized size = 0.75 \begin{align*} - \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}} \end{align*}

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 + 875160*x + 75582)/(1511640*x**20)

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Giac [A]  time = 1.13124, size = 81, normalized size = 1. \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/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 + 875160*x + 75582)/x^20