3.621 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{22}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{1}{10 x^{10}}-\frac{1}{x^{11}}-\frac{55}{12 x^{12}}-\frac{165}{13 x^{13}}-\frac{165}{7 x^{14}}-\frac{154}{5 x^{15}}-\frac{231}{8 x^{16}}-\frac{330}{17 x^{17}}-\frac{55}{6 x^{18}}-\frac{55}{19 x^{19}}-\frac{11}{20 x^{20}}-\frac{1}{21 x^{21}} \]

[Out]

-1/(21*x^21) - 11/(20*x^20) - 55/(19*x^19) - 55/(6*x^18) - 330/(17*x^17) - 231/(8*x^16) - 154/(5*x^15) - 165/(
7*x^14) - 165/(13*x^13) - 55/(12*x^12) - x^(-11) - 1/(10*x^10)

________________________________________________________________________________________

Rubi [A]  time = 0.0216553, antiderivative size = 83, 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} \[ -\frac{1}{10 x^{10}}-\frac{1}{x^{11}}-\frac{55}{12 x^{12}}-\frac{165}{13 x^{13}}-\frac{165}{7 x^{14}}-\frac{154}{5 x^{15}}-\frac{231}{8 x^{16}}-\frac{330}{17 x^{17}}-\frac{55}{6 x^{18}}-\frac{55}{19 x^{19}}-\frac{11}{20 x^{20}}-\frac{1}{21 x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^22,x]

[Out]

-1/(21*x^21) - 11/(20*x^20) - 55/(19*x^19) - 55/(6*x^18) - 330/(17*x^17) - 231/(8*x^16) - 154/(5*x^15) - 165/(
7*x^14) - 165/(13*x^13) - 55/(12*x^12) - x^(-11) - 1/(10*x^10)

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^{22}} \, dx &=\int \frac{(1+x)^{11}}{x^{22}} \, dx\\ &=\int \left (\frac{1}{x^{22}}+\frac{11}{x^{21}}+\frac{55}{x^{20}}+\frac{165}{x^{19}}+\frac{330}{x^{18}}+\frac{462}{x^{17}}+\frac{462}{x^{16}}+\frac{330}{x^{15}}+\frac{165}{x^{14}}+\frac{55}{x^{13}}+\frac{11}{x^{12}}+\frac{1}{x^{11}}\right ) \, dx\\ &=-\frac{1}{21 x^{21}}-\frac{11}{20 x^{20}}-\frac{55}{19 x^{19}}-\frac{55}{6 x^{18}}-\frac{330}{17 x^{17}}-\frac{231}{8 x^{16}}-\frac{154}{5 x^{15}}-\frac{165}{7 x^{14}}-\frac{165}{13 x^{13}}-\frac{55}{12 x^{12}}-\frac{1}{x^{11}}-\frac{1}{10 x^{10}}\\ \end{align*}

Mathematica [A]  time = 0.004206, size = 83, normalized size = 1. \[ -\frac{1}{10 x^{10}}-\frac{1}{x^{11}}-\frac{55}{12 x^{12}}-\frac{165}{13 x^{13}}-\frac{165}{7 x^{14}}-\frac{154}{5 x^{15}}-\frac{231}{8 x^{16}}-\frac{330}{17 x^{17}}-\frac{55}{6 x^{18}}-\frac{55}{19 x^{19}}-\frac{11}{20 x^{20}}-\frac{1}{21 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^22,x]

[Out]

-1/(21*x^21) - 11/(20*x^20) - 55/(19*x^19) - 55/(6*x^18) - 330/(17*x^17) - 231/(8*x^16) - 154/(5*x^15) - 165/(
7*x^14) - 165/(13*x^13) - 55/(12*x^12) - x^(-11) - 1/(10*x^10)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 62, normalized size = 0.8 \begin{align*} -{\frac{1}{21\,{x}^{21}}}-{\frac{11}{20\,{x}^{20}}}-{\frac{55}{19\,{x}^{19}}}-{\frac{55}{6\,{x}^{18}}}-{\frac{330}{17\,{x}^{17}}}-{\frac{231}{8\,{x}^{16}}}-{\frac{154}{5\,{x}^{15}}}-{\frac{165}{7\,{x}^{14}}}-{\frac{165}{13\,{x}^{13}}}-{\frac{55}{12\,{x}^{12}}}-{x}^{-11}-{\frac{1}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)*(x^2+2*x+1)^5/x^22,x)

[Out]

-1/21/x^21-11/20/x^20-55/19/x^19-55/6/x^18-330/17/x^17-231/8/x^16-154/5/x^15-165/7/x^14-165/13/x^13-55/12/x^12
-1/x^11-1/10/x^10

________________________________________________________________________________________

Maxima [A]  time = 1.01334, size = 81, normalized size = 0.98 \begin{align*} -\frac{352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="maxima")

[Out]

-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 83140200*x^7 + 108636528*x^6 + 10184674
5*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21

________________________________________________________________________________________

Fricas [A]  time = 1.19624, size = 255, normalized size = 3.07 \begin{align*} -\frac{352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="fricas")

[Out]

-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 83140200*x^7 + 108636528*x^6 + 10184674
5*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21

________________________________________________________________________________________

Sympy [A]  time = 0.231117, size = 61, normalized size = 0.73 \begin{align*} - \frac{352716 x^{11} + 3527160 x^{10} + 16166150 x^{9} + 44767800 x^{8} + 83140200 x^{7} + 108636528 x^{6} + 101846745 x^{5} + 68468400 x^{4} + 32332300 x^{3} + 10210200 x^{2} + 1939938 x + 167960}{3527160 x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**22,x)

[Out]

-(352716*x**11 + 3527160*x**10 + 16166150*x**9 + 44767800*x**8 + 83140200*x**7 + 108636528*x**6 + 101846745*x*
*5 + 68468400*x**4 + 32332300*x**3 + 10210200*x**2 + 1939938*x + 167960)/(3527160*x**21)

________________________________________________________________________________________

Giac [A]  time = 1.17526, size = 81, normalized size = 0.98 \begin{align*} -\frac{352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="giac")

[Out]

-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 83140200*x^7 + 108636528*x^6 + 10184674
5*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21