3.592 \(\int x^7 (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{19} (x+1)^{19}-\frac{7}{18} (x+1)^{18}+\frac{21}{17} (x+1)^{17}-\frac{35}{16} (x+1)^{16}+\frac{7}{3} (x+1)^{15}-\frac{3}{2} (x+1)^{14}+\frac{7}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(1 + x)^16)/16 + (21*(1 + x)^17
)/17 - (7*(1 + x)^18)/18 + (1 + x)^19/19

________________________________________________________________________________________

Rubi [A]  time = 0.0230896, antiderivative size = 73, 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} \[ \frac{1}{19} (x+1)^{19}-\frac{7}{18} (x+1)^{18}+\frac{21}{17} (x+1)^{17}-\frac{35}{16} (x+1)^{16}+\frac{7}{3} (x+1)^{15}-\frac{3}{2} (x+1)^{14}+\frac{7}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(1 + x)^16)/16 + (21*(1 + x)^17
)/17 - (7*(1 + x)^18)/18 + (1 + x)^19/19

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 x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+7 (1+x)^{12}-21 (1+x)^{13}+35 (1+x)^{14}-35 (1+x)^{15}+21 (1+x)^{16}-7 (1+x)^{17}+(1+x)^{18}\right ) \, dx\\ &=-\frac{1}{12} (1+x)^{12}+\frac{7}{13} (1+x)^{13}-\frac{3}{2} (1+x)^{14}+\frac{7}{3} (1+x)^{15}-\frac{35}{16} (1+x)^{16}+\frac{21}{17} (1+x)^{17}-\frac{7}{18} (1+x)^{18}+\frac{1}{19} (1+x)^{19}\\ \end{align*}

Mathematica [A]  time = 0.0015205, size = 79, normalized size = 1.08 \[ \frac{x^{19}}{19}+\frac{11 x^{18}}{18}+\frac{55 x^{17}}{17}+\frac{165 x^{16}}{16}+22 x^{15}+33 x^{14}+\frac{462 x^{13}}{13}+\frac{55 x^{12}}{2}+15 x^{11}+\frac{11 x^{10}}{2}+\frac{11 x^9}{9}+\frac{x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^14 + 22*x^15 + (165*x^16)/16 +
(55*x^17)/17 + (11*x^18)/18 + x^19/19

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 62, normalized size = 0.9 \begin{align*}{\frac{{x}^{19}}{19}}+{\frac{11\,{x}^{18}}{18}}+{\frac{55\,{x}^{17}}{17}}+{\frac{165\,{x}^{16}}{16}}+22\,{x}^{15}+33\,{x}^{14}+{\frac{462\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{2}}+15\,{x}^{11}+{\frac{11\,{x}^{10}}{2}}+{\frac{11\,{x}^{9}}{9}}+{\frac{{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/19*x^19+11/18*x^18+55/17*x^17+165/16*x^16+22*x^15+33*x^14+462/13*x^13+55/2*x^12+15*x^11+11/2*x^10+11/9*x^9+1
/8*x^8

________________________________________________________________________________________

Maxima [A]  time = 0.976102, size = 82, normalized size = 1.12 \begin{align*} \frac{1}{19} \, x^{19} + \frac{11}{18} \, x^{18} + \frac{55}{17} \, x^{17} + \frac{165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac{462}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 15 \, x^{11} + \frac{11}{2} \, x^{10} + \frac{11}{9} \, x^{9} + \frac{1}{8} \, x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8

________________________________________________________________________________________

Fricas [A]  time = 1.10025, size = 189, normalized size = 2.59 \begin{align*} \frac{1}{19} x^{19} + \frac{11}{18} x^{18} + \frac{55}{17} x^{17} + \frac{165}{16} x^{16} + 22 x^{15} + 33 x^{14} + \frac{462}{13} x^{13} + \frac{55}{2} x^{12} + 15 x^{11} + \frac{11}{2} x^{10} + \frac{11}{9} x^{9} + \frac{1}{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8

________________________________________________________________________________________

Sympy [A]  time = 0.096142, size = 71, normalized size = 0.97 \begin{align*} \frac{x^{19}}{19} + \frac{11 x^{18}}{18} + \frac{55 x^{17}}{17} + \frac{165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac{462 x^{13}}{13} + \frac{55 x^{12}}{2} + 15 x^{11} + \frac{11 x^{10}}{2} + \frac{11 x^{9}}{9} + \frac{x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**19/19 + 11*x**18/18 + 55*x**17/17 + 165*x**16/16 + 22*x**15 + 33*x**14 + 462*x**13/13 + 55*x**12/2 + 15*x**
11 + 11*x**10/2 + 11*x**9/9 + x**8/8

________________________________________________________________________________________

Giac [A]  time = 1.15045, size = 82, normalized size = 1.12 \begin{align*} \frac{1}{19} \, x^{19} + \frac{11}{18} \, x^{18} + \frac{55}{17} \, x^{17} + \frac{165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac{462}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 15 \, x^{11} + \frac{11}{2} \, x^{10} + \frac{11}{9} \, x^{9} + \frac{1}{8} \, x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8