3.595 \(\int x^4 (1+x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{16} (x+1)^{16}-\frac{4}{15} (x+1)^{15}+\frac{3}{7} (x+1)^{14}-\frac{4}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

[Out]

(1 + x)^12/12 - (4*(1 + x)^13)/13 + (3*(1 + x)^14)/7 - (4*(1 + x)^15)/15 + (1 + x)^16/16

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Rubi [A]  time = 0.0177084, antiderivative size = 46, 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} \[ \frac{1}{16} (x+1)^{16}-\frac{4}{15} (x+1)^{15}+\frac{3}{7} (x+1)^{14}-\frac{4}{13} (x+1)^{13}+\frac{1}{12} (x+1)^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12 - (4*(1 + x)^13)/13 + (3*(1 + x)^14)/7 - (4*(1 + x)^15)/15 + (1 + x)^16/16

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^4 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^4 (1+x)^{11} \, dx\\ &=\int \left ((1+x)^{11}-4 (1+x)^{12}+6 (1+x)^{13}-4 (1+x)^{14}+(1+x)^{15}\right ) \, dx\\ &=\frac{1}{12} (1+x)^{12}-\frac{4}{13} (1+x)^{13}+\frac{3}{7} (1+x)^{14}-\frac{4}{15} (1+x)^{15}+\frac{1}{16} (1+x)^{16}\\ \end{align*}

Mathematica [A]  time = 0.0014685, size = 83, normalized size = 1.8 \[ \frac{x^{16}}{16}+\frac{11 x^{15}}{15}+\frac{55 x^{14}}{14}+\frac{165 x^{13}}{13}+\frac{55 x^{12}}{2}+42 x^{11}+\frac{231 x^{10}}{5}+\frac{110 x^9}{3}+\frac{165 x^8}{8}+\frac{55 x^7}{7}+\frac{11 x^6}{6}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^5/5 + (11*x^6)/6 + (55*x^7)/7 + (165*x^8)/8 + (110*x^9)/3 + (231*x^10)/5 + 42*x^11 + (55*x^12)/2 + (165*x^13
)/13 + (55*x^14)/14 + (11*x^15)/15 + x^16/16

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Maple [A]  time = 0., size = 62, normalized size = 1.4 \begin{align*}{\frac{{x}^{16}}{16}}+{\frac{11\,{x}^{15}}{15}}+{\frac{55\,{x}^{14}}{14}}+{\frac{165\,{x}^{13}}{13}}+{\frac{55\,{x}^{12}}{2}}+42\,{x}^{11}+{\frac{231\,{x}^{10}}{5}}+{\frac{110\,{x}^{9}}{3}}+{\frac{165\,{x}^{8}}{8}}+{\frac{55\,{x}^{7}}{7}}+{\frac{11\,{x}^{6}}{6}}+{\frac{{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(1+x)*(x^2+2*x+1)^5,x)

[Out]

1/16*x^16+11/15*x^15+55/14*x^14+165/13*x^13+55/2*x^12+42*x^11+231/5*x^10+110/3*x^9+165/8*x^8+55/7*x^7+11/6*x^6
+1/5*x^5

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Maxima [A]  time = 1.01568, size = 82, normalized size = 1.78 \begin{align*} \frac{1}{16} \, x^{16} + \frac{11}{15} \, x^{15} + \frac{55}{14} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 42 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{165}{8} \, x^{8} + \frac{55}{7} \, x^{7} + \frac{11}{6} \, x^{6} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*x^10 + 110/3*x^9 + 165/8*x^8 +
55/7*x^7 + 11/6*x^6 + 1/5*x^5

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Fricas [A]  time = 1.00039, size = 192, normalized size = 4.17 \begin{align*} \frac{1}{16} x^{16} + \frac{11}{15} x^{15} + \frac{55}{14} x^{14} + \frac{165}{13} x^{13} + \frac{55}{2} x^{12} + 42 x^{11} + \frac{231}{5} x^{10} + \frac{110}{3} x^{9} + \frac{165}{8} x^{8} + \frac{55}{7} x^{7} + \frac{11}{6} x^{6} + \frac{1}{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*x^10 + 110/3*x^9 + 165/8*x^8 +
55/7*x^7 + 11/6*x^6 + 1/5*x^5

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Sympy [B]  time = 0.076351, size = 75, normalized size = 1.63 \begin{align*} \frac{x^{16}}{16} + \frac{11 x^{15}}{15} + \frac{55 x^{14}}{14} + \frac{165 x^{13}}{13} + \frac{55 x^{12}}{2} + 42 x^{11} + \frac{231 x^{10}}{5} + \frac{110 x^{9}}{3} + \frac{165 x^{8}}{8} + \frac{55 x^{7}}{7} + \frac{11 x^{6}}{6} + \frac{x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**16/16 + 11*x**15/15 + 55*x**14/14 + 165*x**13/13 + 55*x**12/2 + 42*x**11 + 231*x**10/5 + 110*x**9/3 + 165*x
**8/8 + 55*x**7/7 + 11*x**6/6 + x**5/5

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Giac [A]  time = 1.13549, size = 82, normalized size = 1.78 \begin{align*} \frac{1}{16} \, x^{16} + \frac{11}{15} \, x^{15} + \frac{55}{14} \, x^{14} + \frac{165}{13} \, x^{13} + \frac{55}{2} \, x^{12} + 42 \, x^{11} + \frac{231}{5} \, x^{10} + \frac{110}{3} \, x^{9} + \frac{165}{8} \, x^{8} + \frac{55}{7} \, x^{7} + \frac{11}{6} \, x^{6} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/16*x^16 + 11/15*x^15 + 55/14*x^14 + 165/13*x^13 + 55/2*x^12 + 42*x^11 + 231/5*x^10 + 110/3*x^9 + 165/8*x^8 +
55/7*x^7 + 11/6*x^6 + 1/5*x^5