3.58 \(\int (8+24 x+8 x^2-15 x^3+8 x^4)^3 \, dx\)

Optimal. Leaf size=76 \[ \frac{512 x^{13}}{13}-240 x^{12}+\frac{6936 x^{11}}{11}-\frac{4527 x^{10}}{10}-\frac{2936 x^9}{3}+2097 x^8+\frac{5528 x^7}{7}-2976 x^6-\frac{384 x^5}{5}+5040 x^4+5120 x^3+2304 x^2+512 x \]

[Out]

512*x + 2304*x^2 + 5120*x^3 + 5040*x^4 - (384*x^5)/5 - 2976*x^6 + (5528*x^7)/7 + 2097*x^8 - (2936*x^9)/3 - (45
27*x^10)/10 + (6936*x^11)/11 - 240*x^12 + (512*x^13)/13

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Rubi [A]  time = 0.023682, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2061} \[ \frac{512 x^{13}}{13}-240 x^{12}+\frac{6936 x^{11}}{11}-\frac{4527 x^{10}}{10}-\frac{2936 x^9}{3}+2097 x^8+\frac{5528 x^7}{7}-2976 x^6-\frac{384 x^5}{5}+5040 x^4+5120 x^3+2304 x^2+512 x \]

Antiderivative was successfully verified.

[In]

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^3,x]

[Out]

512*x + 2304*x^2 + 5120*x^3 + 5040*x^4 - (384*x^5)/5 - 2976*x^6 + (5528*x^7)/7 + 2097*x^8 - (2936*x^9)/3 - (45
27*x^10)/10 + (6936*x^11)/11 - 240*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+24 x+8 x^2-15 x^3+8 x^4\right )^3 \, dx &=\int \left (512+4608 x+15360 x^2+20160 x^3-384 x^4-17856 x^5+5528 x^6+16776 x^7-8808 x^8-4527 x^9+6936 x^{10}-2880 x^{11}+512 x^{12}\right ) \, dx\\ &=512 x+2304 x^2+5120 x^3+5040 x^4-\frac{384 x^5}{5}-2976 x^6+\frac{5528 x^7}{7}+2097 x^8-\frac{2936 x^9}{3}-\frac{4527 x^{10}}{10}+\frac{6936 x^{11}}{11}-240 x^{12}+\frac{512 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0015606, size = 76, normalized size = 1. \[ \frac{512 x^{13}}{13}-240 x^{12}+\frac{6936 x^{11}}{11}-\frac{4527 x^{10}}{10}-\frac{2936 x^9}{3}+2097 x^8+\frac{5528 x^7}{7}-2976 x^6-\frac{384 x^5}{5}+5040 x^4+5120 x^3+2304 x^2+512 x \]

Antiderivative was successfully verified.

[In]

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^3,x]

[Out]

512*x + 2304*x^2 + 5120*x^3 + 5040*x^4 - (384*x^5)/5 - 2976*x^6 + (5528*x^7)/7 + 2097*x^8 - (2936*x^9)/3 - (45
27*x^10)/10 + (6936*x^11)/11 - 240*x^12 + (512*x^13)/13

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Maple [A]  time = 0., size = 65, normalized size = 0.9 \begin{align*} 512\,x+2304\,{x}^{2}+5120\,{x}^{3}+5040\,{x}^{4}-{\frac{384\,{x}^{5}}{5}}-2976\,{x}^{6}+{\frac{5528\,{x}^{7}}{7}}+2097\,{x}^{8}-{\frac{2936\,{x}^{9}}{3}}-{\frac{4527\,{x}^{10}}{10}}+{\frac{6936\,{x}^{11}}{11}}-240\,{x}^{12}+{\frac{512\,{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^4-15*x^3+8*x^2+24*x+8)^3,x)

[Out]

512*x+2304*x^2+5120*x^3+5040*x^4-384/5*x^5-2976*x^6+5528/7*x^7+2097*x^8-2936/3*x^9-4527/10*x^10+6936/11*x^11-2
40*x^12+512/13*x^13

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Maxima [A]  time = 1.1588, size = 86, normalized size = 1.13 \begin{align*} \frac{512}{13} \, x^{13} - 240 \, x^{12} + \frac{6936}{11} \, x^{11} - \frac{4527}{10} \, x^{10} - \frac{2936}{3} \, x^{9} + 2097 \, x^{8} + \frac{5528}{7} \, x^{7} - 2976 \, x^{6} - \frac{384}{5} \, x^{5} + 5040 \, x^{4} + 5120 \, x^{3} + 2304 \, x^{2} + 512 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^3,x, algorithm="maxima")

[Out]

512/13*x^13 - 240*x^12 + 6936/11*x^11 - 4527/10*x^10 - 2936/3*x^9 + 2097*x^8 + 5528/7*x^7 - 2976*x^6 - 384/5*x
^5 + 5040*x^4 + 5120*x^3 + 2304*x^2 + 512*x

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Fricas [A]  time = 1.25592, size = 209, normalized size = 2.75 \begin{align*} \frac{512}{13} x^{13} - 240 x^{12} + \frac{6936}{11} x^{11} - \frac{4527}{10} x^{10} - \frac{2936}{3} x^{9} + 2097 x^{8} + \frac{5528}{7} x^{7} - 2976 x^{6} - \frac{384}{5} x^{5} + 5040 x^{4} + 5120 x^{3} + 2304 x^{2} + 512 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^3,x, algorithm="fricas")

[Out]

512/13*x^13 - 240*x^12 + 6936/11*x^11 - 4527/10*x^10 - 2936/3*x^9 + 2097*x^8 + 5528/7*x^7 - 2976*x^6 - 384/5*x
^5 + 5040*x^4 + 5120*x^3 + 2304*x^2 + 512*x

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Sympy [A]  time = 0.067505, size = 73, normalized size = 0.96 \begin{align*} \frac{512 x^{13}}{13} - 240 x^{12} + \frac{6936 x^{11}}{11} - \frac{4527 x^{10}}{10} - \frac{2936 x^{9}}{3} + 2097 x^{8} + \frac{5528 x^{7}}{7} - 2976 x^{6} - \frac{384 x^{5}}{5} + 5040 x^{4} + 5120 x^{3} + 2304 x^{2} + 512 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**4-15*x**3+8*x**2+24*x+8)**3,x)

[Out]

512*x**13/13 - 240*x**12 + 6936*x**11/11 - 4527*x**10/10 - 2936*x**9/3 + 2097*x**8 + 5528*x**7/7 - 2976*x**6 -
384*x**5/5 + 5040*x**4 + 5120*x**3 + 2304*x**2 + 512*x

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Giac [A]  time = 1.13712, size = 86, normalized size = 1.13 \begin{align*} \frac{512}{13} \, x^{13} - 240 \, x^{12} + \frac{6936}{11} \, x^{11} - \frac{4527}{10} \, x^{10} - \frac{2936}{3} \, x^{9} + 2097 \, x^{8} + \frac{5528}{7} \, x^{7} - 2976 \, x^{6} - \frac{384}{5} \, x^{5} + 5040 \, x^{4} + 5120 \, x^{3} + 2304 \, x^{2} + 512 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^4-15*x^3+8*x^2+24*x+8)^3,x, algorithm="giac")

[Out]

512/13*x^13 - 240*x^12 + 6936/11*x^11 - 4527/10*x^10 - 2936/3*x^9 + 2097*x^8 + 5528/7*x^7 - 2976*x^6 - 384/5*x
^5 + 5040*x^4 + 5120*x^3 + 2304*x^2 + 512*x