3.46 \(\int (8+8 x-x^3+8 x^4)^3 \, dx\)

Optimal. Leaf size=74 \[ \frac{512 x^{13}}{13}-16 x^{12}+\frac{24 x^{11}}{11}+\frac{307 x^{10}}{2}+128 x^9-45 x^8+\frac{1560 x^7}{7}+480 x^6+\frac{1152 x^5}{5}+80 x^4+512 x^3+768 x^2+512 x \]

[Out]

512*x + 768*x^2 + 512*x^3 + 80*x^4 + (1152*x^5)/5 + 480*x^6 + (1560*x^7)/7 - 45*x^8 + 128*x^9 + (307*x^10)/2 +
(24*x^11)/11 - 16*x^12 + (512*x^13)/13

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Rubi [A]  time = 0.0225196, antiderivative size = 74, 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} \[ \frac{512 x^{13}}{13}-16 x^{12}+\frac{24 x^{11}}{11}+\frac{307 x^{10}}{2}+128 x^9-45 x^8+\frac{1560 x^7}{7}+480 x^6+\frac{1152 x^5}{5}+80 x^4+512 x^3+768 x^2+512 x \]

Antiderivative was successfully verified.

[In]

Int[(8 + 8*x - x^3 + 8*x^4)^3,x]

[Out]

512*x + 768*x^2 + 512*x^3 + 80*x^4 + (1152*x^5)/5 + 480*x^6 + (1560*x^7)/7 - 45*x^8 + 128*x^9 + (307*x^10)/2 +
(24*x^11)/11 - 16*x^12 + (512*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 (8+8 x-x^3+8 x^4\right )^3 \, dx &=\int \left (512+1536 x+1536 x^2+320 x^3+1152 x^4+2880 x^5+1560 x^6-360 x^7+1152 x^8+1535 x^9+24 x^{10}-192 x^{11}+512 x^{12}\right ) \, dx\\ &=512 x+768 x^2+512 x^3+80 x^4+\frac{1152 x^5}{5}+480 x^6+\frac{1560 x^7}{7}-45 x^8+128 x^9+\frac{307 x^{10}}{2}+\frac{24 x^{11}}{11}-16 x^{12}+\frac{512 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0012485, size = 74, normalized size = 1. \[ \frac{512 x^{13}}{13}-16 x^{12}+\frac{24 x^{11}}{11}+\frac{307 x^{10}}{2}+128 x^9-45 x^8+\frac{1560 x^7}{7}+480 x^6+\frac{1152 x^5}{5}+80 x^4+512 x^3+768 x^2+512 x \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 8*x - x^3 + 8*x^4)^3,x]

[Out]

512*x + 768*x^2 + 512*x^3 + 80*x^4 + (1152*x^5)/5 + 480*x^6 + (1560*x^7)/7 - 45*x^8 + 128*x^9 + (307*x^10)/2 +
(24*x^11)/11 - 16*x^12 + (512*x^13)/13

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Maple [A]  time = 0.002, size = 65, normalized size = 0.9 \begin{align*} 512\,x+768\,{x}^{2}+512\,{x}^{3}+80\,{x}^{4}+{\frac{1152\,{x}^{5}}{5}}+480\,{x}^{6}+{\frac{1560\,{x}^{7}}{7}}-45\,{x}^{8}+128\,{x}^{9}+{\frac{307\,{x}^{10}}{2}}+{\frac{24\,{x}^{11}}{11}}-16\,{x}^{12}+{\frac{512\,{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^4-x^3+8*x+8)^3,x)

[Out]

512*x+768*x^2+512*x^3+80*x^4+1152/5*x^5+480*x^6+1560/7*x^7-45*x^8+128*x^9+307/2*x^10+24/11*x^11-16*x^12+512/13
*x^13

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Maxima [A]  time = 1.16596, size = 86, normalized size = 1.16 \begin{align*} \frac{512}{13} \, x^{13} - 16 \, x^{12} + \frac{24}{11} \, x^{11} + \frac{307}{2} \, x^{10} + 128 \, x^{9} - 45 \, x^{8} + \frac{1560}{7} \, x^{7} + 480 \, x^{6} + \frac{1152}{5} \, x^{5} + 80 \, x^{4} + 512 \, x^{3} + 768 \, x^{2} + 512 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^3,x, algorithm="maxima")

[Out]

512/13*x^13 - 16*x^12 + 24/11*x^11 + 307/2*x^10 + 128*x^9 - 45*x^8 + 1560/7*x^7 + 480*x^6 + 1152/5*x^5 + 80*x^
4 + 512*x^3 + 768*x^2 + 512*x

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Fricas [A]  time = 1.12793, size = 190, normalized size = 2.57 \begin{align*} \frac{512}{13} x^{13} - 16 x^{12} + \frac{24}{11} x^{11} + \frac{307}{2} x^{10} + 128 x^{9} - 45 x^{8} + \frac{1560}{7} x^{7} + 480 x^{6} + \frac{1152}{5} x^{5} + 80 x^{4} + 512 x^{3} + 768 x^{2} + 512 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^3,x, algorithm="fricas")

[Out]

512/13*x^13 - 16*x^12 + 24/11*x^11 + 307/2*x^10 + 128*x^9 - 45*x^8 + 1560/7*x^7 + 480*x^6 + 1152/5*x^5 + 80*x^
4 + 512*x^3 + 768*x^2 + 512*x

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Sympy [A]  time = 0.064054, size = 71, normalized size = 0.96 \begin{align*} \frac{512 x^{13}}{13} - 16 x^{12} + \frac{24 x^{11}}{11} + \frac{307 x^{10}}{2} + 128 x^{9} - 45 x^{8} + \frac{1560 x^{7}}{7} + 480 x^{6} + \frac{1152 x^{5}}{5} + 80 x^{4} + 512 x^{3} + 768 x^{2} + 512 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**4-x**3+8*x+8)**3,x)

[Out]

512*x**13/13 - 16*x**12 + 24*x**11/11 + 307*x**10/2 + 128*x**9 - 45*x**8 + 1560*x**7/7 + 480*x**6 + 1152*x**5/
5 + 80*x**4 + 512*x**3 + 768*x**2 + 512*x

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Giac [A]  time = 1.11133, size = 86, normalized size = 1.16 \begin{align*} \frac{512}{13} \, x^{13} - 16 \, x^{12} + \frac{24}{11} \, x^{11} + \frac{307}{2} \, x^{10} + 128 \, x^{9} - 45 \, x^{8} + \frac{1560}{7} \, x^{7} + 480 \, x^{6} + \frac{1152}{5} \, x^{5} + 80 \, x^{4} + 512 \, x^{3} + 768 \, x^{2} + 512 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-x^3+8*x+8)^3,x, algorithm="giac")

[Out]

512/13*x^13 - 16*x^12 + 24/11*x^11 + 307/2*x^10 + 128*x^9 - 45*x^8 + 1560/7*x^7 + 480*x^6 + 1152/5*x^5 + 80*x^
4 + 512*x^3 + 768*x^2 + 512*x