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

Optimal. Leaf size=55 \[ \frac{1}{17} (x+1)^{17}-\frac{5}{16} (x+1)^{16}+\frac{2}{3} (x+1)^{15}-\frac{5}{7} (x+1)^{14}+\frac{5}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (5*(1 + x)^13)/13 - (5*(1 + x)^14)/7 + (2*(1 + x)^15)/3 - (5*(1 + x)^16)/16 + (1 + x)^17/17

________________________________________________________________________________________

Rubi [A]  time = 0.0194168, antiderivative size = 55, 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} \[ \frac{1}{17} (x+1)^{17}-\frac{5}{16} (x+1)^{16}+\frac{2}{3} (x+1)^{15}-\frac{5}{7} (x+1)^{14}+\frac{5}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (5*(1 + x)^13)/13 - (5*(1 + x)^14)/7 + (2*(1 + x)^15)/3 - (5*(1 + x)^16)/16 + (1 + x)^17/17

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^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^5 (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+5 (1+x)^{12}-10 (1+x)^{13}+10 (1+x)^{14}-5 (1+x)^{15}+(1+x)^{16}\right ) \, dx\\ &=-\frac{1}{12} (1+x)^{12}+\frac{5}{13} (1+x)^{13}-\frac{5}{7} (1+x)^{14}+\frac{2}{3} (1+x)^{15}-\frac{5}{16} (1+x)^{16}+\frac{1}{17} (1+x)^{17}\\ \end{align*}

Mathematica [A]  time = 0.0018855, size = 81, normalized size = 1.47 \[ \frac{x^{17}}{17}+\frac{11 x^{16}}{16}+\frac{11 x^{15}}{3}+\frac{165 x^{14}}{14}+\frac{330 x^{13}}{13}+\frac{77 x^{12}}{2}+42 x^{11}+33 x^{10}+\frac{55 x^9}{3}+\frac{55 x^8}{8}+\frac{11 x^7}{7}+\frac{x^6}{6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^6/6 + (11*x^7)/7 + (55*x^8)/8 + (55*x^9)/3 + 33*x^10 + 42*x^11 + (77*x^12)/2 + (330*x^13)/13 + (165*x^14)/14
+ (11*x^15)/3 + (11*x^16)/16 + x^17/17

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 62, normalized size = 1.1 \begin{align*}{\frac{{x}^{17}}{17}}+{\frac{11\,{x}^{16}}{16}}+{\frac{11\,{x}^{15}}{3}}+{\frac{165\,{x}^{14}}{14}}+{\frac{330\,{x}^{13}}{13}}+{\frac{77\,{x}^{12}}{2}}+42\,{x}^{11}+33\,{x}^{10}+{\frac{55\,{x}^{9}}{3}}+{\frac{55\,{x}^{8}}{8}}+{\frac{11\,{x}^{7}}{7}}+{\frac{{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17+11/16*x^16+11/3*x^15+165/14*x^14+330/13*x^13+77/2*x^12+42*x^11+33*x^10+55/3*x^9+55/8*x^8+11/7*x^7+1/
6*x^6

________________________________________________________________________________________

Maxima [A]  time = 1.02198, size = 82, normalized size = 1.49 \begin{align*} \frac{1}{17} \, x^{17} + \frac{11}{16} \, x^{16} + \frac{11}{3} \, x^{15} + \frac{165}{14} \, x^{14} + \frac{330}{13} \, x^{13} + \frac{77}{2} \, x^{12} + 42 \, x^{11} + 33 \, x^{10} + \frac{55}{3} \, x^{9} + \frac{55}{8} \, x^{8} + \frac{11}{7} \, x^{7} + \frac{1}{6} \, x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55
/8*x^8 + 11/7*x^7 + 1/6*x^6

________________________________________________________________________________________

Fricas [A]  time = 1.10358, size = 188, normalized size = 3.42 \begin{align*} \frac{1}{17} x^{17} + \frac{11}{16} x^{16} + \frac{11}{3} x^{15} + \frac{165}{14} x^{14} + \frac{330}{13} x^{13} + \frac{77}{2} x^{12} + 42 x^{11} + 33 x^{10} + \frac{55}{3} x^{9} + \frac{55}{8} x^{8} + \frac{11}{7} x^{7} + \frac{1}{6} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55
/8*x^8 + 11/7*x^7 + 1/6*x^6

________________________________________________________________________________________

Sympy [A]  time = 0.082029, size = 73, normalized size = 1.33 \begin{align*} \frac{x^{17}}{17} + \frac{11 x^{16}}{16} + \frac{11 x^{15}}{3} + \frac{165 x^{14}}{14} + \frac{330 x^{13}}{13} + \frac{77 x^{12}}{2} + 42 x^{11} + 33 x^{10} + \frac{55 x^{9}}{3} + \frac{55 x^{8}}{8} + \frac{11 x^{7}}{7} + \frac{x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**17/17 + 11*x**16/16 + 11*x**15/3 + 165*x**14/14 + 330*x**13/13 + 77*x**12/2 + 42*x**11 + 33*x**10 + 55*x**9
/3 + 55*x**8/8 + 11*x**7/7 + x**6/6

________________________________________________________________________________________

Giac [A]  time = 1.14692, size = 82, normalized size = 1.49 \begin{align*} \frac{1}{17} \, x^{17} + \frac{11}{16} \, x^{16} + \frac{11}{3} \, x^{15} + \frac{165}{14} \, x^{14} + \frac{330}{13} \, x^{13} + \frac{77}{2} \, x^{12} + 42 \, x^{11} + 33 \, x^{10} + \frac{55}{3} \, x^{9} + \frac{55}{8} \, x^{8} + \frac{11}{7} \, x^{7} + \frac{1}{6} \, x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55
/8*x^8 + 11/7*x^7 + 1/6*x^6