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+1152/5*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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Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 74, normalized size = 1.00 \[ \frac {512 x^{13}}{13}-16 x^{12}+\frac {24 x^{11}}{11}+\frac {307 x^{10}}{2}+128 x^9-45 x^8+\frac {1560 x^7}{7}+480 x^6+\frac {1152 x^5}{5}+80 x^4+512 x^3+768 x^2+512 x \]

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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fricas [A]  time = 0.35, size = 64, normalized size = 0.86 \[ \frac {512}{13} x^{13} - 16 x^{12} + \frac {24}{11} x^{11} + \frac {307}{2} x^{10} + 128 x^{9} - 45 x^{8} + \frac {1560}{7} x^{7} + 480 x^{6} + \frac {1152}{5} x^{5} + 80 x^{4} + 512 x^{3} + 768 x^{2} + 512 x \]

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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giac [A]  time = 0.33, size = 64, normalized size = 0.86 \[ \frac {512}{13} \, x^{13} - 16 \, x^{12} + \frac {24}{11} \, x^{11} + \frac {307}{2} \, x^{10} + 128 \, x^{9} - 45 \, x^{8} + \frac {1560}{7} \, x^{7} + 480 \, x^{6} + \frac {1152}{5} \, x^{5} + 80 \, x^{4} + 512 \, x^{3} + 768 \, x^{2} + 512 \, x \]

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

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maple [A]  time = 0.00, size = 65, normalized size = 0.88 \[ \frac {512}{13} x^{13}-16 x^{12}+\frac {24}{11} x^{11}+\frac {307}{2} x^{10}+128 x^{9}-45 x^{8}+\frac {1560}{7} x^{7}+480 x^{6}+\frac {1152}{5} x^{5}+80 x^{4}+512 x^{3}+768 x^{2}+512 x \]

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 = 0.94, size = 64, normalized size = 0.86 \[ \frac {512}{13} \, x^{13} - 16 \, x^{12} + \frac {24}{11} \, x^{11} + \frac {307}{2} \, x^{10} + 128 \, x^{9} - 45 \, x^{8} + \frac {1560}{7} \, x^{7} + 480 \, x^{6} + \frac {1152}{5} \, x^{5} + 80 \, x^{4} + 512 \, x^{3} + 768 \, x^{2} + 512 \, x \]

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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mupad [B]  time = 0.08, size = 64, normalized size = 0.86 \[ \frac {512\,x^{13}}{13}-16\,x^{12}+\frac {24\,x^{11}}{11}+\frac {307\,x^{10}}{2}+128\,x^9-45\,x^8+\frac {1560\,x^7}{7}+480\,x^6+\frac {1152\,x^5}{5}+80\,x^4+512\,x^3+768\,x^2+512\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - x^3 + 8*x^4 + 8)^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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sympy [A]  time = 0.07, size = 71, normalized size = 0.96 \[ \frac {512 x^{13}}{13} - 16 x^{12} + \frac {24 x^{11}}{11} + \frac {307 x^{10}}{2} + 128 x^{9} - 45 x^{8} + \frac {1560 x^{7}}{7} + 480 x^{6} + \frac {1152 x^{5}}{5} + 80 x^{4} + 512 x^{3} + 768 x^{2} + 512 x \]

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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