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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0217264, 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} \[ \frac{x^7}{7}+\frac{11 x^6}{6}+11 x^5+\frac{165 x^4}{4}+110 x^3+231 x^2-\frac{55}{2 x^2}-\frac{11}{3 x^3}-\frac{1}{4 x^4}+462 x-\frac{165}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0025982, size = 70, normalized size = 1. \[ \frac{x^7}{7}+\frac{11 x^6}{6}+11 x^5+\frac{165 x^4}{4}+110 x^3+231 x^2-\frac{55}{2 x^2}-\frac{11}{3 x^3}-\frac{1}{4 x^4}+462 x-\frac{165}{x}+330 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01605, size = 78, normalized size = 1.11 \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 - \frac{1980 \, x^{3} + 330 \, x^{2} + 44 \, x + 3}{12 \, x^{4}} + 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^5,x, algorithm="maxima")

[Out]

1/7*x^7 + 11/6*x^6 + 11*x^5 + 165/4*x^4 + 110*x^3 + 231*x^2 + 462*x - 1/12*(1980*x^3 + 330*x^2 + 44*x + 3)/x^4
+ 330*log(x)

________________________________________________________________________________________

Fricas [A]  time = 1.24607, size = 192, normalized size = 2.74 \begin{align*} \frac{12 \, x^{11} + 154 \, x^{10} + 924 \, x^{9} + 3465 \, x^{8} + 9240 \, x^{7} + 19404 \, x^{6} + 38808 \, x^{5} + 27720 \, x^{4} \log \left (x\right ) - 13860 \, x^{3} - 2310 \, x^{2} - 308 \, x - 21}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.122352, size = 61, normalized size = 0.87 \begin{align*} \frac{x^{7}}{7} + \frac{11 x^{6}}{6} + 11 x^{5} + \frac{165 x^{4}}{4} + 110 x^{3} + 231 x^{2} + 462 x + 330 \log{\left (x \right )} - \frac{1980 x^{3} + 330 x^{2} + 44 x + 3}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7/7 + 11*x**6/6 + 11*x**5 + 165*x**4/4 + 110*x**3 + 231*x**2 + 462*x + 330*log(x) - (1980*x**3 + 330*x**2 +
44*x + 3)/(12*x**4)

________________________________________________________________________________________

Giac [A]  time = 1.137, size = 80, normalized size = 1.14 \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 - \frac{1980 \, x^{3} + 330 \, x^{2} + 44 \, x + 3}{12 \, x^{4}} + 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^5,x, algorithm="giac")

[Out]

1/7*x^7 + 11/6*x^6 + 11*x^5 + 165/4*x^4 + 110*x^3 + 231*x^2 + 462*x - 1/12*(1980*x^3 + 330*x^2 + 44*x + 3)/x^4
+ 330*log(abs(x))