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

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

[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]

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Rubi [A]  time = 0.020874, antiderivative size = 70, 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} \[ -\frac{165}{2 x^2}-\frac{110}{x^3}-\frac{231}{2 x^4}-\frac{462}{5 x^5}-\frac{55}{x^6}-\frac{165}{7 x^7}-\frac{55}{8 x^8}-\frac{11}{9 x^9}-\frac{1}{10 x^{10}}+x-\frac{55}{x}+11 \log (x) \]

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*}

Mathematica [A]  time = 0.0031841, size = 70, normalized size = 1. \[ -\frac{165}{2 x^2}-\frac{110}{x^3}-\frac{231}{2 x^4}-\frac{462}{5 x^5}-\frac{55}{x^6}-\frac{165}{7 x^7}-\frac{55}{8 x^8}-\frac{11}{9 x^9}-\frac{1}{10 x^{10}}+x-\frac{55}{x}+11 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((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]

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Maple [A]  time = 0.009, size = 57, normalized size = 0.8 \begin{align*} -{\frac{1}{10\,{x}^{10}}}-{\frac{11}{9\,{x}^{9}}}-{\frac{55}{8\,{x}^{8}}}-{\frac{165}{7\,{x}^{7}}}-55\,{x}^{-6}-{\frac{462}{5\,{x}^{5}}}-{\frac{231}{2\,{x}^{4}}}-110\,{x}^{-3}-{\frac{165}{2\,{x}^{2}}}-55\,{x}^{-1}+x+11\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)*(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 = 1.12422, size = 76, normalized size = 1.09 \begin{align*} 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 (x\right ) \end{align*}

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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Fricas [A]  time = 1.26603, size = 219, normalized size = 3.13 \begin{align*} \frac{2520 \, x^{11} + 27720 \, x^{10} \log \left (x\right ) - 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{align*}

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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Sympy [A]  time = 0.162543, size = 56, normalized size = 0.8 \begin{align*} x + 11 \log{\left (x \right )} - \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{align*}

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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Giac [A]  time = 1.11274, size = 77, normalized size = 1.1 \begin{align*} 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{align*}

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))