3.607 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^8} \, dx\)

Optimal. Leaf size=70 \[ \frac{x^4}{4}+\frac{11 x^3}{3}+\frac{55 x^2}{2}-\frac{231}{x^2}-\frac{110}{x^3}-\frac{165}{4 x^4}-\frac{11}{x^5}-\frac{11}{6 x^6}-\frac{1}{7 x^7}+165 x-\frac{462}{x}+330 \log (x) \]

[Out]

-1/(7*x^7) - 11/(6*x^6) - 11/x^5 - 165/(4*x^4) - 110/x^3 - 231/x^2 - 462/x + 165*x + (55*x^2)/2 + (11*x^3)/3 +
 x^4/4 + 330*Log[x]

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Rubi [A]  time = 0.0212897, 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} \[ \frac{x^4}{4}+\frac{11 x^3}{3}+\frac{55 x^2}{2}-\frac{231}{x^2}-\frac{110}{x^3}-\frac{165}{4 x^4}-\frac{11}{x^5}-\frac{11}{6 x^6}-\frac{1}{7 x^7}+165 x-\frac{462}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

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

Mathematica [A]  time = 0.0025562, size = 70, normalized size = 1. \[ \frac{x^4}{4}+\frac{11 x^3}{3}+\frac{55 x^2}{2}-\frac{231}{x^2}-\frac{110}{x^3}-\frac{165}{4 x^4}-\frac{11}{x^5}-\frac{11}{6 x^6}-\frac{1}{7 x^7}+165 x-\frac{462}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-1/(7*x^7) - 11/(6*x^6) - 11/x^5 - 165/(4*x^4) - 110/x^3 - 231/x^2 - 462/x + 165*x + (55*x^2)/2 + (11*x^3)/3 +
 x^4/4 + 330*Log[x]

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Maple [A]  time = 0.006, size = 59, normalized size = 0.8 \begin{align*} -{\frac{1}{7\,{x}^{7}}}-{\frac{11}{6\,{x}^{6}}}-11\,{x}^{-5}-{\frac{165}{4\,{x}^{4}}}-110\,{x}^{-3}-231\,{x}^{-2}-462\,{x}^{-1}+165\,x+{\frac{55\,{x}^{2}}{2}}+{\frac{11\,{x}^{3}}{3}}+{\frac{{x}^{4}}{4}}+330\,\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^8,x)

[Out]

-1/7/x^7-11/6/x^6-11/x^5-165/4/x^4-110/x^3-231/x^2-462/x+165*x+55/2*x^2+11/3*x^3+1/4*x^4+330*ln(x)

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Maxima [A]  time = 1.00113, size = 78, normalized size = 1.11 \begin{align*} \frac{1}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{55}{2} \, x^{2} + 165 \, x - \frac{38808 \, x^{6} + 19404 \, x^{5} + 9240 \, x^{4} + 3465 \, x^{3} + 924 \, x^{2} + 154 \, x + 12}{84 \, x^{7}} + 330 \, \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^8,x, algorithm="maxima")

[Out]

1/4*x^4 + 11/3*x^3 + 55/2*x^2 + 165*x - 1/84*(38808*x^6 + 19404*x^5 + 9240*x^4 + 3465*x^3 + 924*x^2 + 154*x +
12)/x^7 + 330*log(x)

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Fricas [A]  time = 1.52973, size = 192, normalized size = 2.74 \begin{align*} \frac{21 \, x^{11} + 308 \, x^{10} + 2310 \, x^{9} + 13860 \, x^{8} + 27720 \, x^{7} \log \left (x\right ) - 38808 \, x^{6} - 19404 \, x^{5} - 9240 \, x^{4} - 3465 \, x^{3} - 924 \, x^{2} - 154 \, x - 12}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^8,x, algorithm="fricas")

[Out]

1/84*(21*x^11 + 308*x^10 + 2310*x^9 + 13860*x^8 + 27720*x^7*log(x) - 38808*x^6 - 19404*x^5 - 9240*x^4 - 3465*x
^3 - 924*x^2 - 154*x - 12)/x^7

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Sympy [A]  time = 0.141676, size = 61, normalized size = 0.87 \begin{align*} \frac{x^{4}}{4} + \frac{11 x^{3}}{3} + \frac{55 x^{2}}{2} + 165 x + 330 \log{\left (x \right )} - \frac{38808 x^{6} + 19404 x^{5} + 9240 x^{4} + 3465 x^{3} + 924 x^{2} + 154 x + 12}{84 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**8,x)

[Out]

x**4/4 + 11*x**3/3 + 55*x**2/2 + 165*x + 330*log(x) - (38808*x**6 + 19404*x**5 + 9240*x**4 + 3465*x**3 + 924*x
**2 + 154*x + 12)/(84*x**7)

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Giac [A]  time = 1.13724, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{55}{2} \, x^{2} + 165 \, x - \frac{38808 \, x^{6} + 19404 \, x^{5} + 9240 \, x^{4} + 3465 \, x^{3} + 924 \, x^{2} + 154 \, x + 12}{84 \, x^{7}} + 330 \, \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^8,x, algorithm="giac")

[Out]

1/4*x^4 + 11/3*x^3 + 55/2*x^2 + 165*x - 1/84*(38808*x^6 + 19404*x^5 + 9240*x^4 + 3465*x^3 + 924*x^2 + 154*x +
12)/x^7 + 330*log(abs(x))