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

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

________________________________________________________________________________________

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^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) \end {gather*}

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 = 0.00, size = 70, normalized size = 1.00 \begin {gather*} -\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 {x^2}{2}-\frac {165}{x^2}+11 x-\frac {165}{x}+55 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*1/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]

________________________________________________________________________________________

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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 62, normalized size = 0.89 \begin {gather*} \frac {252 \, x^{11} + 5544 \, x^{10} + 27720 \, x^{9} \log \relax (x) - 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 {gather*}

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 59, normalized size = 0.84 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 59, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{2}+11 x +55 \ln \relax (x )-\frac {165}{x}-\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(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)

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 58, normalized size = 0.83 \begin {gather*} \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 \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^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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 58, normalized size = 0.83 \begin {gather*} 11\,x+55\,\ln \relax (x)+\frac {x^2}{2}-\frac {165\,x^8+165\,x^7+154\,x^6+\frac {231\,x^5}{2}+66\,x^4+\frac {55\,x^3}{2}+\frac {55\,x^2}{7}+\frac {11\,x}{8}+\frac {1}{9}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^10,x)

[Out]

11*x + 55*log(x) + x^2/2 - ((11*x)/8 + (55*x^2)/7 + (55*x^3)/2 + 66*x^4 + (231*x^5)/2 + 154*x^6 + 165*x^7 + 16
5*x^8 + 1/9)/x^9

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 60, normalized size = 0.86 \begin {gather*} \frac {x^{2}}{2} + 11 x + 55 \log {\relax (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}} \end {gather*}

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 - 39
60*x**2 - 693*x - 56)/(504*x**9)

________________________________________________________________________________________