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

Optimal. Leaf size=72 \[ \frac {x^{11}}{11}+\frac {11 x^{10}}{10}+\frac {55 x^9}{9}+\frac {165 x^8}{8}+\frac {330 x^7}{7}+77 x^6+\frac {462 x^5}{5}+\frac {165 x^4}{2}+55 x^3+\frac {55 x^2}{2}+11 x+\log (x) \]

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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^{11}}{11}+\frac {11 x^{10}}{10}+\frac {55 x^9}{9}+\frac {165 x^8}{8}+\frac {330 x^7}{7}+77 x^6+\frac {462 x^5}{5}+\frac {165 x^4}{2}+55 x^3+\frac {55 x^2}{2}+11 x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x,x]

[Out]

11*x + (55*x^2)/2 + 55*x^3 + (165*x^4)/2 + (462*x^5)/5 + 77*x^6 + (330*x^7)/7 + (165*x^8)/8 + (55*x^9)/9 + (11
*x^10)/10 + x^11/11 + 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} \, dx &=\int \frac {(1+x)^{11}}{x} \, dx\\ &=\int \left (11+\frac {1}{x}+55 x+165 x^2+330 x^3+462 x^4+462 x^5+330 x^6+165 x^7+55 x^8+11 x^9+x^{10}\right ) \, dx\\ &=11 x+\frac {55 x^2}{2}+55 x^3+\frac {165 x^4}{2}+\frac {462 x^5}{5}+77 x^6+\frac {330 x^7}{7}+\frac {165 x^8}{8}+\frac {55 x^9}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 96, normalized size = 1.33 \begin {gather*} \frac {1}{11} (x+1)^{11}+\frac {1}{10} (x+1)^{10}+\frac {1}{9} (x+1)^9+\frac {1}{8} (x+1)^8+\frac {1}{7} (x+1)^7+\frac {1}{6} (x+1)^6+\frac {1}{5} (x+1)^5+\frac {1}{4} (x+1)^4+\frac {1}{3} (x+1)^3+\frac {1}{2} (x+1)^2+x+\log (-x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 56, normalized size = 0.78 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \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,x, algorithm="fricas")

[Out]

1/11*x^11 + 11/10*x^10 + 55/9*x^9 + 165/8*x^8 + 330/7*x^7 + 77*x^6 + 462/5*x^5 + 165/2*x^4 + 55*x^3 + 55/2*x^2
+ 11*x + log(x)

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giac [A]  time = 0.15, size = 57, normalized size = 0.79 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \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,x, algorithm="giac")

[Out]

1/11*x^11 + 11/10*x^10 + 55/9*x^9 + 165/8*x^8 + 330/7*x^7 + 77*x^6 + 462/5*x^5 + 165/2*x^4 + 55*x^3 + 55/2*x^2
+ 11*x + log(abs(x))

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maple [A]  time = 0.05, size = 57, normalized size = 0.79 \begin {gather*} \frac {x^{11}}{11}+\frac {11 x^{10}}{10}+\frac {55 x^{9}}{9}+\frac {165 x^{8}}{8}+\frac {330 x^{7}}{7}+77 x^{6}+\frac {462 x^{5}}{5}+\frac {165 x^{4}}{2}+55 x^{3}+\frac {55 x^{2}}{2}+11 x +\ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11*x+55/2*x^2+55*x^3+165/2*x^4+462/5*x^5+77*x^6+330/7*x^7+165/8*x^8+55/9*x^9+11/10*x^10+1/11*x^11+ln(x)

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maxima [A]  time = 0.63, size = 56, normalized size = 0.78 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \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,x, algorithm="maxima")

[Out]

1/11*x^11 + 11/10*x^10 + 55/9*x^9 + 165/8*x^8 + 330/7*x^7 + 77*x^6 + 462/5*x^5 + 165/2*x^4 + 55*x^3 + 55/2*x^2
+ 11*x + log(x)

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mupad [B]  time = 0.06, size = 56, normalized size = 0.78 \begin {gather*} 11\,x+\ln \relax (x)+\frac {55\,x^2}{2}+55\,x^3+\frac {165\,x^4}{2}+\frac {462\,x^5}{5}+77\,x^6+\frac {330\,x^7}{7}+\frac {165\,x^8}{8}+\frac {55\,x^9}{9}+\frac {11\,x^{10}}{10}+\frac {x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x,x)

[Out]

11*x + log(x) + (55*x^2)/2 + 55*x^3 + (165*x^4)/2 + (462*x^5)/5 + 77*x^6 + (330*x^7)/7 + (165*x^8)/8 + (55*x^9
)/9 + (11*x^10)/10 + x^11/11

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sympy [A]  time = 0.11, size = 68, normalized size = 0.94 \begin {gather*} \frac {x^{11}}{11} + \frac {11 x^{10}}{10} + \frac {55 x^{9}}{9} + \frac {165 x^{8}}{8} + \frac {330 x^{7}}{7} + 77 x^{6} + \frac {462 x^{5}}{5} + \frac {165 x^{4}}{2} + 55 x^{3} + \frac {55 x^{2}}{2} + 11 x + \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,x)

[Out]

x**11/11 + 11*x**10/10 + 55*x**9/9 + 165*x**8/8 + 330*x**7/7 + 77*x**6 + 462*x**5/5 + 165*x**4/2 + 55*x**3 + 5
5*x**2/2 + 11*x + log(x)

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