3.52 \(\int (1+4 x+4 x^2+4 x^4)^3 \, dx\)

Optimal. Leaf size=69 \[ \frac{64 x^{13}}{13}+\frac{192 x^{11}}{11}+\frac{96 x^{10}}{5}+\frac{80 x^9}{3}+48 x^8+\frac{352 x^7}{7}+48 x^6+\frac{252 x^5}{5}+40 x^4+20 x^3+6 x^2+x \]

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*x^13)/13

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Rubi [A]  time = 0.0200803, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2061} \[ \frac{64 x^{13}}{13}+\frac{192 x^{11}}{11}+\frac{96 x^{10}}{5}+\frac{80 x^9}{3}+48 x^8+\frac{352 x^7}{7}+48 x^6+\frac{252 x^5}{5}+40 x^4+20 x^3+6 x^2+x \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^3,x]

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*x^13)/13

Rule 2061

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (1+4 x+4 x^2+4 x^4\right )^3 \, dx &=\int \left (1+12 x+60 x^2+160 x^3+252 x^4+288 x^5+352 x^6+384 x^7+240 x^8+192 x^9+192 x^{10}+64 x^{12}\right ) \, dx\\ &=x+6 x^2+20 x^3+40 x^4+\frac{252 x^5}{5}+48 x^6+\frac{352 x^7}{7}+48 x^8+\frac{80 x^9}{3}+\frac{96 x^{10}}{5}+\frac{192 x^{11}}{11}+\frac{64 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0009704, size = 69, normalized size = 1. \[ \frac{64 x^{13}}{13}+\frac{192 x^{11}}{11}+\frac{96 x^{10}}{5}+\frac{80 x^9}{3}+48 x^8+\frac{352 x^7}{7}+48 x^6+\frac{252 x^5}{5}+40 x^4+20 x^3+6 x^2+x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^3,x]

[Out]

x + 6*x^2 + 20*x^3 + 40*x^4 + (252*x^5)/5 + 48*x^6 + (352*x^7)/7 + 48*x^8 + (80*x^9)/3 + (96*x^10)/5 + (192*x^
11)/11 + (64*x^13)/13

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Maple [A]  time = 0., size = 58, normalized size = 0.8 \begin{align*} x+6\,{x}^{2}+20\,{x}^{3}+40\,{x}^{4}+{\frac{252\,{x}^{5}}{5}}+48\,{x}^{6}+{\frac{352\,{x}^{7}}{7}}+48\,{x}^{8}+{\frac{80\,{x}^{9}}{3}}+{\frac{96\,{x}^{10}}{5}}+{\frac{192\,{x}^{11}}{11}}+{\frac{64\,{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+4*x^2+4*x+1)^3,x)

[Out]

x+6*x^2+20*x^3+40*x^4+252/5*x^5+48*x^6+352/7*x^7+48*x^8+80/3*x^9+96/5*x^10+192/11*x^11+64/13*x^13

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Maxima [A]  time = 1.15995, size = 77, normalized size = 1.12 \begin{align*} \frac{64}{13} \, x^{13} + \frac{192}{11} \, x^{11} + \frac{96}{5} \, x^{10} + \frac{80}{3} \, x^{9} + 48 \, x^{8} + \frac{352}{7} \, x^{7} + 48 \, x^{6} + \frac{252}{5} \, x^{5} + 40 \, x^{4} + 20 \, x^{3} + 6 \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="maxima")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x

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Fricas [A]  time = 1.12574, size = 163, normalized size = 2.36 \begin{align*} \frac{64}{13} x^{13} + \frac{192}{11} x^{11} + \frac{96}{5} x^{10} + \frac{80}{3} x^{9} + 48 x^{8} + \frac{352}{7} x^{7} + 48 x^{6} + \frac{252}{5} x^{5} + 40 x^{4} + 20 x^{3} + 6 x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="fricas")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x

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Sympy [A]  time = 0.065145, size = 66, normalized size = 0.96 \begin{align*} \frac{64 x^{13}}{13} + \frac{192 x^{11}}{11} + \frac{96 x^{10}}{5} + \frac{80 x^{9}}{3} + 48 x^{8} + \frac{352 x^{7}}{7} + 48 x^{6} + \frac{252 x^{5}}{5} + 40 x^{4} + 20 x^{3} + 6 x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4+4*x**2+4*x+1)**3,x)

[Out]

64*x**13/13 + 192*x**11/11 + 96*x**10/5 + 80*x**9/3 + 48*x**8 + 352*x**7/7 + 48*x**6 + 252*x**5/5 + 40*x**4 +
20*x**3 + 6*x**2 + x

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Giac [A]  time = 1.12538, size = 77, normalized size = 1.12 \begin{align*} \frac{64}{13} \, x^{13} + \frac{192}{11} \, x^{11} + \frac{96}{5} \, x^{10} + \frac{80}{3} \, x^{9} + 48 \, x^{8} + \frac{352}{7} \, x^{7} + 48 \, x^{6} + \frac{252}{5} \, x^{5} + 40 \, x^{4} + 20 \, x^{3} + 6 \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+4*x^2+4*x+1)^3,x, algorithm="giac")

[Out]

64/13*x^13 + 192/11*x^11 + 96/5*x^10 + 80/3*x^9 + 48*x^8 + 352/7*x^7 + 48*x^6 + 252/5*x^5 + 40*x^4 + 20*x^3 +
6*x^2 + x