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

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

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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 {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) \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 70, normalized size = 1.00 \begin {gather*} -\frac {1}{7 x^7}-\frac {11}{6 x^6}-\frac {11}{x^5}+\frac {x^4}{4}-\frac {165}{4 x^4}+\frac {11 x^3}{3}-\frac {110}{x^3}+\frac {55 x^2}{2}-\frac {231}{x^2}+165 x-\frac {462}{x}+330 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/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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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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 62, normalized size = 0.89 \begin {gather*} \frac {21 \, x^{11} + 308 \, x^{10} + 2310 \, x^{9} + 13860 \, x^{8} + 27720 \, x^{7} \log \relax (x) - 38808 \, x^{6} - 19404 \, x^{5} - 9240 \, x^{4} - 3465 \, x^{3} - 924 \, x^{2} - 154 \, x - 12}{84 \, x^{7}} \end {gather*}

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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giac [A]  time = 0.15, size = 59, normalized size = 0.84 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.05, size = 59, normalized size = 0.84 \begin {gather*} \frac {x^{4}}{4}+\frac {11 x^{3}}{3}+\frac {55 x^{2}}{2}+165 x +330 \ln \relax (x )-\frac {462}{x}-\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(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 = 0.48, size = 58, normalized size = 0.83 \begin {gather*} \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 \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^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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mupad [B]  time = 0.03, size = 58, normalized size = 0.83 \begin {gather*} 165\,x+330\,\ln \relax (x)-\frac {462\,x^6+231\,x^5+110\,x^4+\frac {165\,x^3}{4}+11\,x^2+\frac {11\,x}{6}+\frac {1}{7}}{x^7}+\frac {55\,x^2}{2}+\frac {11\,x^3}{3}+\frac {x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^8,x)

[Out]

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

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sympy [A]  time = 0.15, size = 63, normalized size = 0.90 \begin {gather*} \frac {x^{4}}{4} + \frac {11 x^{3}}{3} + \frac {55 x^{2}}{2} + 165 x + 330 \log {\relax (x )} + \frac {- 38808 x^{6} - 19404 x^{5} - 9240 x^{4} - 3465 x^{3} - 924 x^{2} - 154 x - 12}{84 x^{7}} \end {gather*}

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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