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

Optimal. Leaf size=28 \[ \frac{1}{14} (x+1)^{14}-\frac{2}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

[Out]

(1 + x)^12/12 - (2*(1 + x)^13)/13 + (1 + x)^14/14

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Rubi [A]  time = 0.0138082, antiderivative size = 28, 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} \[ \frac{1}{14} (x+1)^{14}-\frac{2}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12 - (2*(1 + x)^13)/13 + (1 + x)^14/14

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

Mathematica [B]  time = 0.001437, size = 79, normalized size = 2.82 \[ \frac{x^{14}}{14}+\frac{11 x^{13}}{13}+\frac{55 x^{12}}{12}+15 x^{11}+33 x^{10}+\frac{154 x^9}{3}+\frac{231 x^8}{4}+\frac{330 x^7}{7}+\frac{55 x^6}{2}+11 x^5+\frac{11 x^4}{4}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/3 + (11*x^4)/4 + 11*x^5 + (55*x^6)/2 + (330*x^7)/7 + (231*x^8)/4 + (154*x^9)/3 + 33*x^10 + 15*x^11 + (55*x
^12)/12 + (11*x^13)/13 + x^14/14

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Maple [B]  time = 0.001, size = 62, normalized size = 2.2 \begin{align*}{\frac{{x}^{14}}{14}}+{\frac{11\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{12}}+15\,{x}^{11}+33\,{x}^{10}+{\frac{154\,{x}^{9}}{3}}+{\frac{231\,{x}^{8}}{4}}+{\frac{330\,{x}^{7}}{7}}+{\frac{55\,{x}^{6}}{2}}+11\,{x}^{5}+{\frac{11\,{x}^{4}}{4}}+{\frac{{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14+11/13*x^13+55/12*x^12+15*x^11+33*x^10+154/3*x^9+231/4*x^8+330/7*x^7+55/2*x^6+11*x^5+11/4*x^4+1/3*x^3

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Maxima [B]  time = 0.957947, size = 82, normalized size = 2.93 \begin{align*} \frac{1}{14} \, x^{14} + \frac{11}{13} \, x^{13} + \frac{55}{12} \, x^{12} + 15 \, x^{11} + 33 \, x^{10} + \frac{154}{3} \, x^{9} + \frac{231}{4} \, x^{8} + \frac{330}{7} \, x^{7} + \frac{55}{2} \, x^{6} + 11 \, x^{5} + \frac{11}{4} \, x^{4} + \frac{1}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14 + 11/13*x^13 + 55/12*x^12 + 15*x^11 + 33*x^10 + 154/3*x^9 + 231/4*x^8 + 330/7*x^7 + 55/2*x^6 + 11*x^
5 + 11/4*x^4 + 1/3*x^3

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Fricas [B]  time = 1.13238, size = 181, normalized size = 6.46 \begin{align*} \frac{1}{14} x^{14} + \frac{11}{13} x^{13} + \frac{55}{12} x^{12} + 15 x^{11} + 33 x^{10} + \frac{154}{3} x^{9} + \frac{231}{4} x^{8} + \frac{330}{7} x^{7} + \frac{55}{2} x^{6} + 11 x^{5} + \frac{11}{4} x^{4} + \frac{1}{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14 + 11/13*x^13 + 55/12*x^12 + 15*x^11 + 33*x^10 + 154/3*x^9 + 231/4*x^8 + 330/7*x^7 + 55/2*x^6 + 11*x^
5 + 11/4*x^4 + 1/3*x^3

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Sympy [B]  time = 0.072574, size = 71, normalized size = 2.54 \begin{align*} \frac{x^{14}}{14} + \frac{11 x^{13}}{13} + \frac{55 x^{12}}{12} + 15 x^{11} + 33 x^{10} + \frac{154 x^{9}}{3} + \frac{231 x^{8}}{4} + \frac{330 x^{7}}{7} + \frac{55 x^{6}}{2} + 11 x^{5} + \frac{11 x^{4}}{4} + \frac{x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**14/14 + 11*x**13/13 + 55*x**12/12 + 15*x**11 + 33*x**10 + 154*x**9/3 + 231*x**8/4 + 330*x**7/7 + 55*x**6/2
+ 11*x**5 + 11*x**4/4 + x**3/3

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Giac [B]  time = 1.11318, size = 82, normalized size = 2.93 \begin{align*} \frac{1}{14} \, x^{14} + \frac{11}{13} \, x^{13} + \frac{55}{12} \, x^{12} + 15 \, x^{11} + 33 \, x^{10} + \frac{154}{3} \, x^{9} + \frac{231}{4} \, x^{8} + \frac{330}{7} \, x^{7} + \frac{55}{2} \, x^{6} + 11 \, x^{5} + \frac{11}{4} \, x^{4} + \frac{1}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14 + 11/13*x^13 + 55/12*x^12 + 15*x^11 + 33*x^10 + 154/3*x^9 + 231/4*x^8 + 330/7*x^7 + 55/2*x^6 + 11*x^
5 + 11/4*x^4 + 1/3*x^3