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

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

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Rubi [A]  time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 83, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*1/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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 60, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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

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giac [A]  time = 0.16, size = 60, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.05, size = 62, normalized size = 0.75 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(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

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maxima [A]  time = 0.57, size = 60, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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

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mupad [B]  time = 1.07, size = 58, normalized size = 0.70 \begin {gather*} -\frac {\frac {x^{11}}{10}+x^{10}+\frac {55\,x^9}{12}+\frac {165\,x^8}{13}+\frac {165\,x^7}{7}+\frac {154\,x^6}{5}+\frac {231\,x^5}{8}+\frac {330\,x^4}{17}+\frac {55\,x^3}{6}+\frac {55\,x^2}{19}+\frac {11\,x}{20}+\frac {1}{21}}{x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((11*x)/20 + (55*x^2)/19 + (55*x^3)/6 + (330*x^4)/17 + (231*x^5)/8 + (154*x^6)/5 + (165*x^7)/7 + (165*x^8)/13
+ (55*x^9)/12 + x^10 + x^11/10 + 1/21)/x^21

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sympy [A]  time = 0.22, size = 61, normalized size = 0.73 \begin {gather*} \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 {gather*}

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)

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