3.611 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{12}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{55}{2 x^2}-\frac{55}{x^3}-\frac{165}{2 x^4}-\frac{462}{5 x^5}-\frac{77}{x^6}-\frac{330}{7 x^7}-\frac{165}{8 x^8}-\frac{55}{9 x^9}-\frac{11}{10 x^{10}}-\frac{1}{11 x^{11}}-\frac{11}{x}+\log (x) \]

[Out]

-1/(11*x^11) - 11/(10*x^10) - 55/(9*x^9) - 165/(8*x^8) - 330/(7*x^7) - 77/x^6 - 462/(5*x^5) - 165/(2*x^4) - 55
/x^3 - 55/(2*x^2) - 11/x + Log[x]

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Rubi [A]  time = 0.0211485, antiderivative size = 74, 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} \[ -\frac{55}{2 x^2}-\frac{55}{x^3}-\frac{165}{2 x^4}-\frac{462}{5 x^5}-\frac{77}{x^6}-\frac{330}{7 x^7}-\frac{165}{8 x^8}-\frac{55}{9 x^9}-\frac{11}{10 x^{10}}-\frac{1}{11 x^{11}}-\frac{11}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-1/(11*x^11) - 11/(10*x^10) - 55/(9*x^9) - 165/(8*x^8) - 330/(7*x^7) - 77/x^6 - 462/(5*x^5) - 165/(2*x^4) - 55
/x^3 - 55/(2*x^2) - 11/x + 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^{12}} \, dx &=\int \frac{(1+x)^{11}}{x^{12}} \, dx\\ &=\int \left (\frac{1}{x^{12}}+\frac{11}{x^{11}}+\frac{55}{x^{10}}+\frac{165}{x^9}+\frac{330}{x^8}+\frac{462}{x^7}+\frac{462}{x^6}+\frac{330}{x^5}+\frac{165}{x^4}+\frac{55}{x^3}+\frac{11}{x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{1}{11 x^{11}}-\frac{11}{10 x^{10}}-\frac{55}{9 x^9}-\frac{165}{8 x^8}-\frac{330}{7 x^7}-\frac{77}{x^6}-\frac{462}{5 x^5}-\frac{165}{2 x^4}-\frac{55}{x^3}-\frac{55}{2 x^2}-\frac{11}{x}+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0025088, size = 74, normalized size = 1. \[ -\frac{55}{2 x^2}-\frac{55}{x^3}-\frac{165}{2 x^4}-\frac{462}{5 x^5}-\frac{77}{x^6}-\frac{330}{7 x^7}-\frac{165}{8 x^8}-\frac{55}{9 x^9}-\frac{11}{10 x^{10}}-\frac{1}{11 x^{11}}-\frac{11}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-1/(11*x^11) - 11/(10*x^10) - 55/(9*x^9) - 165/(8*x^8) - 330/(7*x^7) - 77/x^6 - 462/(5*x^5) - 165/(2*x^4) - 55
/x^3 - 55/(2*x^2) - 11/x + Log[x]

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Maple [A]  time = 0.007, size = 59, normalized size = 0.8 \begin{align*} -{\frac{1}{11\,{x}^{11}}}-{\frac{11}{10\,{x}^{10}}}-{\frac{55}{9\,{x}^{9}}}-{\frac{165}{8\,{x}^{8}}}-{\frac{330}{7\,{x}^{7}}}-77\,{x}^{-6}-{\frac{462}{5\,{x}^{5}}}-{\frac{165}{2\,{x}^{4}}}-55\,{x}^{-3}-{\frac{55}{2\,{x}^{2}}}-11\,{x}^{-1}+\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^12,x)

[Out]

-1/11/x^11-11/10/x^10-55/9/x^9-165/8/x^8-330/7/x^7-77/x^6-462/5/x^5-165/2/x^4-55/x^3-55/2/x^2-11/x+ln(x)

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Maxima [A]  time = 0.999884, size = 78, normalized size = 1.05 \begin{align*} -\frac{304920 \, x^{10} + 762300 \, x^{9} + 1524600 \, x^{8} + 2286900 \, x^{7} + 2561328 \, x^{6} + 2134440 \, x^{5} + 1306800 \, x^{4} + 571725 \, x^{3} + 169400 \, x^{2} + 30492 \, x + 2520}{27720 \, x^{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^12,x, algorithm="maxima")

[Out]

-1/27720*(304920*x^10 + 762300*x^9 + 1524600*x^8 + 2286900*x^7 + 2561328*x^6 + 2134440*x^5 + 1306800*x^4 + 571
725*x^3 + 169400*x^2 + 30492*x + 2520)/x^11 + log(x)

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Fricas [A]  time = 1.24185, size = 235, normalized size = 3.18 \begin{align*} \frac{27720 \, x^{11} \log \left (x\right ) - 304920 \, x^{10} - 762300 \, x^{9} - 1524600 \, x^{8} - 2286900 \, x^{7} - 2561328 \, x^{6} - 2134440 \, x^{5} - 1306800 \, x^{4} - 571725 \, x^{3} - 169400 \, x^{2} - 30492 \, x - 2520}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^12,x, algorithm="fricas")

[Out]

1/27720*(27720*x^11*log(x) - 304920*x^10 - 762300*x^9 - 1524600*x^8 - 2286900*x^7 - 2561328*x^6 - 2134440*x^5
- 1306800*x^4 - 571725*x^3 - 169400*x^2 - 30492*x - 2520)/x^11

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Sympy [A]  time = 0.176887, size = 58, normalized size = 0.78 \begin{align*} \log{\left (x \right )} - \frac{304920 x^{10} + 762300 x^{9} + 1524600 x^{8} + 2286900 x^{7} + 2561328 x^{6} + 2134440 x^{5} + 1306800 x^{4} + 571725 x^{3} + 169400 x^{2} + 30492 x + 2520}{27720 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**12,x)

[Out]

log(x) - (304920*x**10 + 762300*x**9 + 1524600*x**8 + 2286900*x**7 + 2561328*x**6 + 2134440*x**5 + 1306800*x**
4 + 571725*x**3 + 169400*x**2 + 30492*x + 2520)/(27720*x**11)

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Giac [A]  time = 1.158, size = 80, normalized size = 1.08 \begin{align*} -\frac{304920 \, x^{10} + 762300 \, x^{9} + 1524600 \, x^{8} + 2286900 \, x^{7} + 2561328 \, x^{6} + 2134440 \, x^{5} + 1306800 \, x^{4} + 571725 \, x^{3} + 169400 \, x^{2} + 30492 \, x + 2520}{27720 \, x^{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^12,x, algorithm="giac")

[Out]

-1/27720*(304920*x^10 + 762300*x^9 + 1524600*x^8 + 2286900*x^7 + 2561328*x^6 + 2134440*x^5 + 1306800*x^4 + 571
725*x^3 + 169400*x^2 + 30492*x + 2520)/x^11 + log(abs(x))