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

Optimal. Leaf size=19 \[ \frac{1}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (1 + x)^13/13

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Rubi [A]  time = 0.0054111, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {27, 43} \[ \frac{1}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (1 + x)^13/13

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 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+(1+x)^{12}\right ) \, dx\\ &=-\frac{1}{12} (1+x)^{12}+\frac{1}{13} (1+x)^{13}\\ \end{align*}

Mathematica [B]  time = 0.0012607, size = 77, normalized size = 4.05 \[ \frac{x^{13}}{13}+\frac{11 x^{12}}{12}+5 x^{11}+\frac{33 x^{10}}{2}+\frac{110 x^9}{3}+\frac{231 x^8}{4}+66 x^7+55 x^6+33 x^5+\frac{55 x^4}{4}+\frac{11 x^3}{3}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 + (11*x^3)/3 + (55*x^4)/4 + 33*x^5 + 55*x^6 + 66*x^7 + (231*x^8)/4 + (110*x^9)/3 + (33*x^10)/2 + 5*x^11
+ (11*x^12)/12 + x^13/13

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Maple [B]  time = 0.001, size = 62, normalized size = 3.3 \begin{align*}{\frac{{x}^{13}}{13}}+{\frac{11\,{x}^{12}}{12}}+5\,{x}^{11}+{\frac{33\,{x}^{10}}{2}}+{\frac{110\,{x}^{9}}{3}}+{\frac{231\,{x}^{8}}{4}}+66\,{x}^{7}+55\,{x}^{6}+33\,{x}^{5}+{\frac{55\,{x}^{4}}{4}}+{\frac{11\,{x}^{3}}{3}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13+11/12*x^12+5*x^11+33/2*x^10+110/3*x^9+231/4*x^8+66*x^7+55*x^6+33*x^5+55/4*x^4+11/3*x^3+1/2*x^2

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Maxima [B]  time = 1.00792, size = 82, normalized size = 4.32 \begin{align*} \frac{1}{13} \, x^{13} + \frac{11}{12} \, x^{12} + 5 \, x^{11} + \frac{33}{2} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{231}{4} \, x^{8} + 66 \, x^{7} + 55 \, x^{6} + 33 \, x^{5} + \frac{55}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13 + 11/12*x^12 + 5*x^11 + 33/2*x^10 + 110/3*x^9 + 231/4*x^8 + 66*x^7 + 55*x^6 + 33*x^5 + 55/4*x^4 + 11
/3*x^3 + 1/2*x^2

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Fricas [B]  time = 1.13056, size = 173, normalized size = 9.11 \begin{align*} \frac{1}{13} x^{13} + \frac{11}{12} x^{12} + 5 x^{11} + \frac{33}{2} x^{10} + \frac{110}{3} x^{9} + \frac{231}{4} x^{8} + 66 x^{7} + 55 x^{6} + 33 x^{5} + \frac{55}{4} x^{4} + \frac{11}{3} x^{3} + \frac{1}{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13 + 11/12*x^12 + 5*x^11 + 33/2*x^10 + 110/3*x^9 + 231/4*x^8 + 66*x^7 + 55*x^6 + 33*x^5 + 55/4*x^4 + 11
/3*x^3 + 1/2*x^2

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Sympy [B]  time = 0.07217, size = 70, normalized size = 3.68 \begin{align*} \frac{x^{13}}{13} + \frac{11 x^{12}}{12} + 5 x^{11} + \frac{33 x^{10}}{2} + \frac{110 x^{9}}{3} + \frac{231 x^{8}}{4} + 66 x^{7} + 55 x^{6} + 33 x^{5} + \frac{55 x^{4}}{4} + \frac{11 x^{3}}{3} + \frac{x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**13/13 + 11*x**12/12 + 5*x**11 + 33*x**10/2 + 110*x**9/3 + 231*x**8/4 + 66*x**7 + 55*x**6 + 33*x**5 + 55*x**
4/4 + 11*x**3/3 + x**2/2

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Giac [B]  time = 1.1657, size = 82, normalized size = 4.32 \begin{align*} \frac{1}{13} \, x^{13} + \frac{11}{12} \, x^{12} + 5 \, x^{11} + \frac{33}{2} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{231}{4} \, x^{8} + 66 \, x^{7} + 55 \, x^{6} + 33 \, x^{5} + \frac{55}{4} \, x^{4} + \frac{11}{3} \, x^{3} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13 + 11/12*x^12 + 5*x^11 + 33/2*x^10 + 110/3*x^9 + 231/4*x^8 + 66*x^7 + 55*x^6 + 33*x^5 + 55/4*x^4 + 11
/3*x^3 + 1/2*x^2