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

Optimal. Leaf size=72 \[ \frac {x^{10}}{10}+\frac {11 x^9}{9}+\frac {55 x^8}{8}+\frac {165 x^7}{7}+55 x^6+\frac {462 x^5}{5}+\frac {231 x^4}{2}+110 x^3+\frac {165 x^2}{2}+55 x-\frac {1}{x}+11 \log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 72, 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^{10}}{10}+\frac {11 x^9}{9}+\frac {55 x^8}{8}+\frac {165 x^7}{7}+55 x^6+\frac {462 x^5}{5}+\frac {231 x^4}{2}+110 x^3+\frac {165 x^2}{2}+55 x-\frac {1}{x}+11 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 55*x + (165*x^2)/2 + 110*x^3 + (231*x^4)/2 + (462*x^5)/5 + 55*x^6 + (165*x^7)/7 + (55*x^8)/8 + (11*x
^9)/9 + x^10/10 + 11*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^2} \, dx &=\int \frac {(1+x)^{11}}{x^2} \, dx\\ &=\int \left (55+\frac {1}{x^2}+\frac {11}{x}+165 x+330 x^2+462 x^3+462 x^4+330 x^5+165 x^6+55 x^7+11 x^8+x^9\right ) \, dx\\ &=-\frac {1}{x}+55 x+\frac {165 x^2}{2}+110 x^3+\frac {231 x^4}{2}+\frac {462 x^5}{5}+55 x^6+\frac {165 x^7}{7}+\frac {55 x^8}{8}+\frac {11 x^9}{9}+\frac {x^{10}}{10}+11 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 72, normalized size = 1.00 \begin {gather*} \frac {x^{10}}{10}+\frac {11 x^9}{9}+\frac {55 x^8}{8}+\frac {165 x^7}{7}+55 x^6+\frac {462 x^5}{5}+\frac {231 x^4}{2}+110 x^3+\frac {165 x^2}{2}+55 x-\frac {1}{x}+11 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 55*x + (165*x^2)/2 + 110*x^3 + (231*x^4)/2 + (462*x^5)/5 + 55*x^6 + (165*x^7)/7 + (55*x^8)/8 + (11*x
^9)/9 + x^10/10 + 11*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^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 62, normalized size = 0.86 \begin {gather*} \frac {252 \, x^{11} + 3080 \, x^{10} + 17325 \, x^{9} + 59400 \, x^{8} + 138600 \, x^{7} + 232848 \, x^{6} + 291060 \, x^{5} + 277200 \, x^{4} + 207900 \, x^{3} + 138600 \, x^{2} + 27720 \, x \log \relax (x) - 2520}{2520 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(252*x^11 + 3080*x^10 + 17325*x^9 + 59400*x^8 + 138600*x^7 + 232848*x^6 + 291060*x^5 + 277200*x^4 + 207
900*x^3 + 138600*x^2 + 27720*x*log(x) - 2520)/x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 59, normalized size = 0.82 \begin {gather*} \frac {1}{10} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {55}{8} \, x^{8} + \frac {165}{7} \, x^{7} + 55 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {231}{2} \, x^{4} + 110 \, x^{3} + \frac {165}{2} \, x^{2} + 55 \, x - \frac {1}{x} + 11 \, \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^2,x, algorithm="giac")

[Out]

1/10*x^10 + 11/9*x^9 + 55/8*x^8 + 165/7*x^7 + 55*x^6 + 462/5*x^5 + 231/2*x^4 + 110*x^3 + 165/2*x^2 + 55*x - 1/
x + 11*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 59, normalized size = 0.82 \begin {gather*} \frac {x^{10}}{10}+\frac {11 x^{9}}{9}+\frac {55 x^{8}}{8}+\frac {165 x^{7}}{7}+55 x^{6}+\frac {462 x^{5}}{5}+\frac {231 x^{4}}{2}+110 x^{3}+\frac {165 x^{2}}{2}+55 x +11 \ln \relax (x )-\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x+55*x+165/2*x^2+110*x^3+231/2*x^4+462/5*x^5+55*x^6+165/7*x^7+55/8*x^8+11/9*x^9+1/10*x^10+11*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 58, normalized size = 0.81 \begin {gather*} \frac {1}{10} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {55}{8} \, x^{8} + \frac {165}{7} \, x^{7} + 55 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {231}{2} \, x^{4} + 110 \, x^{3} + \frac {165}{2} \, x^{2} + 55 \, x - \frac {1}{x} + 11 \, \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^2,x, algorithm="maxima")

[Out]

1/10*x^10 + 11/9*x^9 + 55/8*x^8 + 165/7*x^7 + 55*x^6 + 462/5*x^5 + 231/2*x^4 + 110*x^3 + 165/2*x^2 + 55*x - 1/
x + 11*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 58, normalized size = 0.81 \begin {gather*} 55\,x+11\,\ln \relax (x)-\frac {1}{x}+\frac {165\,x^2}{2}+110\,x^3+\frac {231\,x^4}{2}+\frac {462\,x^5}{5}+55\,x^6+\frac {165\,x^7}{7}+\frac {55\,x^8}{8}+\frac {11\,x^9}{9}+\frac {x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

55*x + 11*log(x) - 1/x + (165*x^2)/2 + 110*x^3 + (231*x^4)/2 + (462*x^5)/5 + 55*x^6 + (165*x^7)/7 + (55*x^8)/8
+ (11*x^9)/9 + x^10/10

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 66, normalized size = 0.92 \begin {gather*} \frac {x^{10}}{10} + \frac {11 x^{9}}{9} + \frac {55 x^{8}}{8} + \frac {165 x^{7}}{7} + 55 x^{6} + \frac {462 x^{5}}{5} + \frac {231 x^{4}}{2} + 110 x^{3} + \frac {165 x^{2}}{2} + 55 x + 11 \log {\relax (x )} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**10/10 + 11*x**9/9 + 55*x**8/8 + 165*x**7/7 + 55*x**6 + 462*x**5/5 + 231*x**4/2 + 110*x**3 + 165*x**2/2 + 55
*x + 11*log(x) - 1/x

________________________________________________________________________________________