3.609 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{10}} \, dx\)

Optimal. Leaf size=70 \[ \frac{x^2}{2}-\frac{165}{x^2}-\frac{154}{x^3}-\frac{231}{2 x^4}-\frac{66}{x^5}-\frac{55}{2 x^6}-\frac{55}{7 x^7}-\frac{11}{8 x^8}-\frac{1}{9 x^9}+11 x-\frac{165}{x}+55 \log (x) \]

[Out]

-1/(9*x^9) - 11/(8*x^8) - 55/(7*x^7) - 55/(2*x^6) - 66/x^5 - 231/(2*x^4) - 154/x^3 - 165/x^2 - 165/x + 11*x +
x^2/2 + 55*Log[x]

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Rubi [A]  time = 0.0221587, 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} \[ \frac{x^2}{2}-\frac{165}{x^2}-\frac{154}{x^3}-\frac{231}{2 x^4}-\frac{66}{x^5}-\frac{55}{2 x^6}-\frac{55}{7 x^7}-\frac{11}{8 x^8}-\frac{1}{9 x^9}+11 x-\frac{165}{x}+55 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

-1/(9*x^9) - 11/(8*x^8) - 55/(7*x^7) - 55/(2*x^6) - 66/x^5 - 231/(2*x^4) - 154/x^3 - 165/x^2 - 165/x + 11*x +
x^2/2 + 55*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^{10}} \, dx &=\int \frac{(1+x)^{11}}{x^{10}} \, dx\\ &=\int \left (11+\frac{1}{x^{10}}+\frac{11}{x^9}+\frac{55}{x^8}+\frac{165}{x^7}+\frac{330}{x^6}+\frac{462}{x^5}+\frac{462}{x^4}+\frac{330}{x^3}+\frac{165}{x^2}+\frac{55}{x}+x\right ) \, dx\\ &=-\frac{1}{9 x^9}-\frac{11}{8 x^8}-\frac{55}{7 x^7}-\frac{55}{2 x^6}-\frac{66}{x^5}-\frac{231}{2 x^4}-\frac{154}{x^3}-\frac{165}{x^2}-\frac{165}{x}+11 x+\frac{x^2}{2}+55 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0025497, size = 70, normalized size = 1. \[ \frac{x^2}{2}-\frac{165}{x^2}-\frac{154}{x^3}-\frac{231}{2 x^4}-\frac{66}{x^5}-\frac{55}{2 x^6}-\frac{55}{7 x^7}-\frac{11}{8 x^8}-\frac{1}{9 x^9}+11 x-\frac{165}{x}+55 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

-1/(9*x^9) - 11/(8*x^8) - 55/(7*x^7) - 55/(2*x^6) - 66/x^5 - 231/(2*x^4) - 154/x^3 - 165/x^2 - 165/x + 11*x +
x^2/2 + 55*Log[x]

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Maple [A]  time = 0.005, size = 59, normalized size = 0.8 \begin{align*} -{\frac{1}{9\,{x}^{9}}}-{\frac{11}{8\,{x}^{8}}}-{\frac{55}{7\,{x}^{7}}}-{\frac{55}{2\,{x}^{6}}}-66\,{x}^{-5}-{\frac{231}{2\,{x}^{4}}}-154\,{x}^{-3}-165\,{x}^{-2}-165\,{x}^{-1}+11\,x+{\frac{{x}^{2}}{2}}+55\,\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^10,x)

[Out]

-1/9/x^9-11/8/x^8-55/7/x^7-55/2/x^6-66/x^5-231/2/x^4-154/x^3-165/x^2-165/x+11*x+1/2*x^2+55*ln(x)

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Maxima [A]  time = 1.06099, size = 78, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} + 11 \, x - \frac{83160 \, x^{8} + 83160 \, x^{7} + 77616 \, x^{6} + 58212 \, x^{5} + 33264 \, x^{4} + 13860 \, x^{3} + 3960 \, x^{2} + 693 \, x + 56}{504 \, x^{9}} + 55 \, \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^10,x, algorithm="maxima")

[Out]

1/2*x^2 + 11*x - 1/504*(83160*x^8 + 83160*x^7 + 77616*x^6 + 58212*x^5 + 33264*x^4 + 13860*x^3 + 3960*x^2 + 693
*x + 56)/x^9 + 55*log(x)

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Fricas [A]  time = 1.28418, size = 201, normalized size = 2.87 \begin{align*} \frac{252 \, x^{11} + 5544 \, x^{10} + 27720 \, x^{9} \log \left (x\right ) - 83160 \, x^{8} - 83160 \, x^{7} - 77616 \, x^{6} - 58212 \, x^{5} - 33264 \, x^{4} - 13860 \, x^{3} - 3960 \, x^{2} - 693 \, x - 56}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^10,x, algorithm="fricas")

[Out]

1/504*(252*x^11 + 5544*x^10 + 27720*x^9*log(x) - 83160*x^8 - 83160*x^7 - 77616*x^6 - 58212*x^5 - 33264*x^4 - 1
3860*x^3 - 3960*x^2 - 693*x - 56)/x^9

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Sympy [A]  time = 0.156132, size = 58, normalized size = 0.83 \begin{align*} \frac{x^{2}}{2} + 11 x + 55 \log{\left (x \right )} - \frac{83160 x^{8} + 83160 x^{7} + 77616 x^{6} + 58212 x^{5} + 33264 x^{4} + 13860 x^{3} + 3960 x^{2} + 693 x + 56}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**10,x)

[Out]

x**2/2 + 11*x + 55*log(x) - (83160*x**8 + 83160*x**7 + 77616*x**6 + 58212*x**5 + 33264*x**4 + 13860*x**3 + 396
0*x**2 + 693*x + 56)/(504*x**9)

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Giac [A]  time = 1.12616, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{2} \, x^{2} + 11 \, x - \frac{83160 \, x^{8} + 83160 \, x^{7} + 77616 \, x^{6} + 58212 \, x^{5} + 33264 \, x^{4} + 13860 \, x^{3} + 3960 \, x^{2} + 693 \, x + 56}{504 \, x^{9}} + 55 \, \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^10,x, algorithm="giac")

[Out]

1/2*x^2 + 11*x - 1/504*(83160*x^8 + 83160*x^7 + 77616*x^6 + 58212*x^5 + 33264*x^4 + 13860*x^3 + 3960*x^2 + 693
*x + 56)/x^9 + 55*log(abs(x))