3.6.37 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^{11}} \, dx\)

Optimal. Leaf size=70 \[ -\frac {1}{10 x^{10}}-\frac {11}{9 x^9}-\frac {55}{8 x^8}-\frac {165}{7 x^7}-\frac {55}{x^6}-\frac {462}{5 x^5}-\frac {231}{2 x^4}-\frac {110}{x^3}-\frac {165}{2 x^2}+x-\frac {55}{x}+11 \log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 70, 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 {165}{2 x^2}-\frac {110}{x^3}-\frac {231}{2 x^4}-\frac {462}{5 x^5}-\frac {55}{x^6}-\frac {165}{7 x^7}-\frac {55}{8 x^8}-\frac {11}{9 x^9}-\frac {1}{10 x^{10}}+x-\frac {55}{x}+11 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^11,x]

[Out]

-1/(10*x^10) - 11/(9*x^9) - 55/(8*x^8) - 165/(7*x^7) - 55/x^6 - 462/(5*x^5) - 231/(2*x^4) - 110/x^3 - 165/(2*x
^2) - 55/x + x + 11*Log[x]

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^{11}} \, dx &=\int \frac {(1+x)^{11}}{x^{11}} \, dx\\ &=\int \left (1+\frac {1}{x^{11}}+\frac {11}{x^{10}}+\frac {55}{x^9}+\frac {165}{x^8}+\frac {330}{x^7}+\frac {462}{x^6}+\frac {462}{x^5}+\frac {330}{x^4}+\frac {165}{x^3}+\frac {55}{x^2}+\frac {11}{x}\right ) \, dx\\ &=-\frac {1}{10 x^{10}}-\frac {11}{9 x^9}-\frac {55}{8 x^8}-\frac {165}{7 x^7}-\frac {55}{x^6}-\frac {462}{5 x^5}-\frac {231}{2 x^4}-\frac {110}{x^3}-\frac {165}{2 x^2}-\frac {55}{x}+x+11 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 70, normalized size = 1.00 \begin {gather*} -\frac {1}{10 x^{10}}-\frac {11}{9 x^9}-\frac {55}{8 x^8}-\frac {165}{7 x^7}-\frac {55}{x^6}-\frac {462}{5 x^5}-\frac {231}{2 x^4}-\frac {110}{x^3}-\frac {165}{2 x^2}+x-\frac {55}{x}+11 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^11,x]

[Out]

-1/10*1/x^10 - 11/(9*x^9) - 55/(8*x^8) - 165/(7*x^7) - 55/x^6 - 462/(5*x^5) - 231/(2*x^4) - 110/x^3 - 165/(2*x
^2) - 55/x + x + 11*Log[x]

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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^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^11,x]

[Out]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^11, x]

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fricas [A]  time = 0.40, size = 62, normalized size = 0.89 \begin {gather*} \frac {2520 \, x^{11} + 27720 \, x^{10} \log \relax (x) - 138600 \, x^{9} - 207900 \, x^{8} - 277200 \, x^{7} - 291060 \, x^{6} - 232848 \, x^{5} - 138600 \, x^{4} - 59400 \, x^{3} - 17325 \, x^{2} - 3080 \, x - 252}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*x^11 + 27720*x^10*log(x) - 138600*x^9 - 207900*x^8 - 277200*x^7 - 291060*x^6 - 232848*x^5 - 13860
0*x^4 - 59400*x^3 - 17325*x^2 - 3080*x - 252)/x^10

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giac [A]  time = 0.15, size = 57, normalized size = 0.81 \begin {gather*} x - \frac {138600 \, x^{9} + 207900 \, x^{8} + 277200 \, x^{7} + 291060 \, x^{6} + 232848 \, x^{5} + 138600 \, x^{4} + 59400 \, x^{3} + 17325 \, x^{2} + 3080 \, x + 252}{2520 \, x^{10}} + 11 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^11,x, algorithm="giac")

[Out]

x - 1/2520*(138600*x^9 + 207900*x^8 + 277200*x^7 + 291060*x^6 + 232848*x^5 + 138600*x^4 + 59400*x^3 + 17325*x^
2 + 3080*x + 252)/x^10 + 11*log(abs(x))

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maple [A]  time = 0.05, size = 57, normalized size = 0.81 \begin {gather*} x +11 \ln \relax (x )-\frac {55}{x}-\frac {165}{2 x^{2}}-\frac {110}{x^{3}}-\frac {231}{2 x^{4}}-\frac {462}{5 x^{5}}-\frac {55}{x^{6}}-\frac {165}{7 x^{7}}-\frac {55}{8 x^{8}}-\frac {11}{9 x^{9}}-\frac {1}{10 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^11,x)

[Out]

-1/10/x^10-11/9/x^9-55/8/x^8-165/7/x^7-55/x^6-462/5/x^5-231/2/x^4-110/x^3-165/2/x^2-55/x+x+11*ln(x)

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maxima [A]  time = 0.53, size = 56, normalized size = 0.80 \begin {gather*} x - \frac {138600 \, x^{9} + 207900 \, x^{8} + 277200 \, x^{7} + 291060 \, x^{6} + 232848 \, x^{5} + 138600 \, x^{4} + 59400 \, x^{3} + 17325 \, x^{2} + 3080 \, x + 252}{2520 \, x^{10}} + 11 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^11,x, algorithm="maxima")

[Out]

x - 1/2520*(138600*x^9 + 207900*x^8 + 277200*x^7 + 291060*x^6 + 232848*x^5 + 138600*x^4 + 59400*x^3 + 17325*x^
2 + 3080*x + 252)/x^10 + 11*log(x)

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mupad [B]  time = 1.08, size = 62, normalized size = 0.89 \begin {gather*} -\frac {\frac {11\,x}{9}-11\,x^{10}\,\ln \relax (x)+\frac {55\,x^2}{8}+\frac {165\,x^3}{7}+55\,x^4+\frac {462\,x^5}{5}+\frac {231\,x^6}{2}+110\,x^7+\frac {165\,x^8}{2}+55\,x^9-x^{11}+\frac {1}{10}}{x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^11,x)

[Out]

-((11*x)/9 - 11*x^10*log(x) + (55*x^2)/8 + (165*x^3)/7 + 55*x^4 + (462*x^5)/5 + (231*x^6)/2 + 110*x^7 + (165*x
^8)/2 + 55*x^9 - x^11 + 1/10)/x^10

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sympy [A]  time = 0.17, size = 58, normalized size = 0.83 \begin {gather*} x + 11 \log {\relax (x )} + \frac {- 138600 x^{9} - 207900 x^{8} - 277200 x^{7} - 291060 x^{6} - 232848 x^{5} - 138600 x^{4} - 59400 x^{3} - 17325 x^{2} - 3080 x - 252}{2520 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**11,x)

[Out]

x + 11*log(x) + (-138600*x**9 - 207900*x**8 - 277200*x**7 - 291060*x**6 - 232848*x**5 - 138600*x**4 - 59400*x*
*3 - 17325*x**2 - 3080*x - 252)/(2520*x**10)

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