3.596 \(\int x^3 (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{15} (x+1)^{15}-\frac{3}{14} (x+1)^{14}+\frac{3}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

[Out]

-(1 + x)^12/12 + (3*(1 + x)^13)/13 - (3*(1 + x)^14)/14 + (1 + x)^15/15

________________________________________________________________________________________

Rubi [A]  time = 0.0162581, antiderivative size = 37, 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} \[ \frac{1}{15} (x+1)^{15}-\frac{3}{14} (x+1)^{14}+\frac{3}{13} (x+1)^{13}-\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-(1 + x)^12/12 + (3*(1 + x)^13)/13 - (3*(1 + x)^14)/14 + (1 + x)^15/15

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

Mathematica [B]  time = 0.0014113, size = 83, normalized size = 2.24 \[ \frac{x^{15}}{15}+\frac{11 x^{14}}{14}+\frac{55 x^{13}}{13}+\frac{55 x^{12}}{4}+30 x^{11}+\frac{231 x^{10}}{5}+\frac{154 x^9}{3}+\frac{165 x^8}{4}+\frac{165 x^7}{7}+\frac{55 x^6}{6}+\frac{11 x^5}{5}+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^4/4 + (11*x^5)/5 + (55*x^6)/6 + (165*x^7)/7 + (165*x^8)/4 + (154*x^9)/3 + (231*x^10)/5 + 30*x^11 + (55*x^12)
/4 + (55*x^13)/13 + (11*x^14)/14 + x^15/15

________________________________________________________________________________________

Maple [B]  time = 0.001, size = 62, normalized size = 1.7 \begin{align*}{\frac{{x}^{15}}{15}}+{\frac{11\,{x}^{14}}{14}}+{\frac{55\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{4}}+30\,{x}^{11}+{\frac{231\,{x}^{10}}{5}}+{\frac{154\,{x}^{9}}{3}}+{\frac{165\,{x}^{8}}{4}}+{\frac{165\,{x}^{7}}{7}}+{\frac{55\,{x}^{6}}{6}}+{\frac{11\,{x}^{5}}{5}}+{\frac{{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/15*x^15+11/14*x^14+55/13*x^13+55/4*x^12+30*x^11+231/5*x^10+154/3*x^9+165/4*x^8+165/7*x^7+55/6*x^6+11/5*x^5+1
/4*x^4

________________________________________________________________________________________

Maxima [B]  time = 1.00283, size = 82, normalized size = 2.22 \begin{align*} \frac{1}{15} \, x^{15} + \frac{11}{14} \, x^{14} + \frac{55}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 30 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{154}{3} \, x^{9} + \frac{165}{4} \, x^{8} + \frac{165}{7} \, x^{7} + \frac{55}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{1}{4} \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x^9 + 165/4*x^8 + 165/7*x^7 + 5
5/6*x^6 + 11/5*x^5 + 1/4*x^4

________________________________________________________________________________________

Fricas [B]  time = 1.15406, size = 189, normalized size = 5.11 \begin{align*} \frac{1}{15} x^{15} + \frac{11}{14} x^{14} + \frac{55}{13} x^{13} + \frac{55}{4} x^{12} + 30 x^{11} + \frac{231}{5} x^{10} + \frac{154}{3} x^{9} + \frac{165}{4} x^{8} + \frac{165}{7} x^{7} + \frac{55}{6} x^{6} + \frac{11}{5} x^{5} + \frac{1}{4} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x^9 + 165/4*x^8 + 165/7*x^7 + 5
5/6*x^6 + 11/5*x^5 + 1/4*x^4

________________________________________________________________________________________

Sympy [B]  time = 0.078233, size = 75, normalized size = 2.03 \begin{align*} \frac{x^{15}}{15} + \frac{11 x^{14}}{14} + \frac{55 x^{13}}{13} + \frac{55 x^{12}}{4} + 30 x^{11} + \frac{231 x^{10}}{5} + \frac{154 x^{9}}{3} + \frac{165 x^{8}}{4} + \frac{165 x^{7}}{7} + \frac{55 x^{6}}{6} + \frac{11 x^{5}}{5} + \frac{x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**15/15 + 11*x**14/14 + 55*x**13/13 + 55*x**12/4 + 30*x**11 + 231*x**10/5 + 154*x**9/3 + 165*x**8/4 + 165*x**
7/7 + 55*x**6/6 + 11*x**5/5 + x**4/4

________________________________________________________________________________________

Giac [B]  time = 1.11783, size = 82, normalized size = 2.22 \begin{align*} \frac{1}{15} \, x^{15} + \frac{11}{14} \, x^{14} + \frac{55}{13} \, x^{13} + \frac{55}{4} \, x^{12} + 30 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{154}{3} \, x^{9} + \frac{165}{4} \, x^{8} + \frac{165}{7} \, x^{7} + \frac{55}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{1}{4} \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/15*x^15 + 11/14*x^14 + 55/13*x^13 + 55/4*x^12 + 30*x^11 + 231/5*x^10 + 154/3*x^9 + 165/4*x^8 + 165/7*x^7 + 5
5/6*x^6 + 11/5*x^5 + 1/4*x^4