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

Optimal. Leaf size=96 \[ \frac{4096 x^{17}}{17}-128 x^{16}+\frac{128 x^{15}}{5}+1168 x^{14}+\frac{10241 x^{13}}{13}-448 x^{12}+\frac{25312 x^{11}}{11}+\frac{21488 x^{10}}{5}+1408 x^9+1376 x^8+6784 x^7+7168 x^6+\frac{14336 x^5}{5}+3584 x^4+8192 x^3+8192 x^2+4096 x \]

[Out]

4096*x + 8192*x^2 + 8192*x^3 + 3584*x^4 + (14336*x^5)/5 + 7168*x^6 + 6784*x^7 + 1376*x^8 + 1408*x^9 + (21488*x
^10)/5 + (25312*x^11)/11 - 448*x^12 + (10241*x^13)/13 + 1168*x^14 + (128*x^15)/5 - 128*x^16 + (4096*x^17)/17

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Rubi [A]  time = 0.0301725, antiderivative size = 96, 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} \[ \frac{4096 x^{17}}{17}-128 x^{16}+\frac{128 x^{15}}{5}+1168 x^{14}+\frac{10241 x^{13}}{13}-448 x^{12}+\frac{25312 x^{11}}{11}+\frac{21488 x^{10}}{5}+1408 x^9+1376 x^8+6784 x^7+7168 x^6+\frac{14336 x^5}{5}+3584 x^4+8192 x^3+8192 x^2+4096 x \]

Antiderivative was successfully verified.

[In]

Int[(8 + 8*x - x^3 + 8*x^4)^4,x]

[Out]

4096*x + 8192*x^2 + 8192*x^3 + 3584*x^4 + (14336*x^5)/5 + 7168*x^6 + 6784*x^7 + 1376*x^8 + 1408*x^9 + (21488*x
^10)/5 + (25312*x^11)/11 - 448*x^12 + (10241*x^13)/13 + 1168*x^14 + (128*x^15)/5 - 128*x^16 + (4096*x^17)/17

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 )^4 \, dx &=\int \left (4096+16384 x+24576 x^2+14336 x^3+14336 x^4+43008 x^5+47488 x^6+11008 x^7+12672 x^8+42976 x^9+25312 x^{10}-5376 x^{11}+10241 x^{12}+16352 x^{13}+384 x^{14}-2048 x^{15}+4096 x^{16}\right ) \, dx\\ &=4096 x+8192 x^2+8192 x^3+3584 x^4+\frac{14336 x^5}{5}+7168 x^6+6784 x^7+1376 x^8+1408 x^9+\frac{21488 x^{10}}{5}+\frac{25312 x^{11}}{11}-448 x^{12}+\frac{10241 x^{13}}{13}+1168 x^{14}+\frac{128 x^{15}}{5}-128 x^{16}+\frac{4096 x^{17}}{17}\\ \end{align*}

Mathematica [A]  time = 0.0015055, size = 96, normalized size = 1. \[ \frac{4096 x^{17}}{17}-128 x^{16}+\frac{128 x^{15}}{5}+1168 x^{14}+\frac{10241 x^{13}}{13}-448 x^{12}+\frac{25312 x^{11}}{11}+\frac{21488 x^{10}}{5}+1408 x^9+1376 x^8+6784 x^7+7168 x^6+\frac{14336 x^5}{5}+3584 x^4+8192 x^3+8192 x^2+4096 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

4096*x + 8192*x^2 + 8192*x^3 + 3584*x^4 + (14336*x^5)/5 + 7168*x^6 + 6784*x^7 + 1376*x^8 + 1408*x^9 + (21488*x
^10)/5 + (25312*x^11)/11 - 448*x^12 + (10241*x^13)/13 + 1168*x^14 + (128*x^15)/5 - 128*x^16 + (4096*x^17)/17

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Maple [A]  time = 0., size = 85, normalized size = 0.9 \begin{align*} 4096\,x+8192\,{x}^{2}+8192\,{x}^{3}+3584\,{x}^{4}+{\frac{14336\,{x}^{5}}{5}}+7168\,{x}^{6}+6784\,{x}^{7}+1376\,{x}^{8}+1408\,{x}^{9}+{\frac{21488\,{x}^{10}}{5}}+{\frac{25312\,{x}^{11}}{11}}-448\,{x}^{12}+{\frac{10241\,{x}^{13}}{13}}+1168\,{x}^{14}+{\frac{128\,{x}^{15}}{5}}-128\,{x}^{16}+{\frac{4096\,{x}^{17}}{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096*x+8192*x^2+8192*x^3+3584*x^4+14336/5*x^5+7168*x^6+6784*x^7+1376*x^8+1408*x^9+21488/5*x^10+25312/11*x^11-4
48*x^12+10241/13*x^13+1168*x^14+128/5*x^15-128*x^16+4096/17*x^17

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Maxima [A]  time = 1.04722, size = 113, normalized size = 1.18 \begin{align*} \frac{4096}{17} \, x^{17} - 128 \, x^{16} + \frac{128}{5} \, x^{15} + 1168 \, x^{14} + \frac{10241}{13} \, x^{13} - 448 \, x^{12} + \frac{25312}{11} \, x^{11} + \frac{21488}{5} \, x^{10} + 1408 \, x^{9} + 1376 \, x^{8} + 6784 \, x^{7} + 7168 \, x^{6} + \frac{14336}{5} \, x^{5} + 3584 \, x^{4} + 8192 \, x^{3} + 8192 \, x^{2} + 4096 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096/17*x^17 - 128*x^16 + 128/5*x^15 + 1168*x^14 + 10241/13*x^13 - 448*x^12 + 25312/11*x^11 + 21488/5*x^10 + 1
408*x^9 + 1376*x^8 + 6784*x^7 + 7168*x^6 + 14336/5*x^5 + 3584*x^4 + 8192*x^3 + 8192*x^2 + 4096*x

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Fricas [A]  time = 1.11129, size = 281, normalized size = 2.93 \begin{align*} \frac{4096}{17} x^{17} - 128 x^{16} + \frac{128}{5} x^{15} + 1168 x^{14} + \frac{10241}{13} x^{13} - 448 x^{12} + \frac{25312}{11} x^{11} + \frac{21488}{5} x^{10} + 1408 x^{9} + 1376 x^{8} + 6784 x^{7} + 7168 x^{6} + \frac{14336}{5} x^{5} + 3584 x^{4} + 8192 x^{3} + 8192 x^{2} + 4096 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096/17*x^17 - 128*x^16 + 128/5*x^15 + 1168*x^14 + 10241/13*x^13 - 448*x^12 + 25312/11*x^11 + 21488/5*x^10 + 1
408*x^9 + 1376*x^8 + 6784*x^7 + 7168*x^6 + 14336/5*x^5 + 3584*x^4 + 8192*x^3 + 8192*x^2 + 4096*x

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Sympy [A]  time = 0.073529, size = 94, normalized size = 0.98 \begin{align*} \frac{4096 x^{17}}{17} - 128 x^{16} + \frac{128 x^{15}}{5} + 1168 x^{14} + \frac{10241 x^{13}}{13} - 448 x^{12} + \frac{25312 x^{11}}{11} + \frac{21488 x^{10}}{5} + 1408 x^{9} + 1376 x^{8} + 6784 x^{7} + 7168 x^{6} + \frac{14336 x^{5}}{5} + 3584 x^{4} + 8192 x^{3} + 8192 x^{2} + 4096 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096*x**17/17 - 128*x**16 + 128*x**15/5 + 1168*x**14 + 10241*x**13/13 - 448*x**12 + 25312*x**11/11 + 21488*x**
10/5 + 1408*x**9 + 1376*x**8 + 6784*x**7 + 7168*x**6 + 14336*x**5/5 + 3584*x**4 + 8192*x**3 + 8192*x**2 + 4096
*x

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Giac [A]  time = 1.16385, size = 113, normalized size = 1.18 \begin{align*} \frac{4096}{17} \, x^{17} - 128 \, x^{16} + \frac{128}{5} \, x^{15} + 1168 \, x^{14} + \frac{10241}{13} \, x^{13} - 448 \, x^{12} + \frac{25312}{11} \, x^{11} + \frac{21488}{5} \, x^{10} + 1408 \, x^{9} + 1376 \, x^{8} + 6784 \, x^{7} + 7168 \, x^{6} + \frac{14336}{5} \, x^{5} + 3584 \, x^{4} + 8192 \, x^{3} + 8192 \, x^{2} + 4096 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096/17*x^17 - 128*x^16 + 128/5*x^15 + 1168*x^14 + 10241/13*x^13 - 448*x^12 + 25312/11*x^11 + 21488/5*x^10 + 1
408*x^9 + 1376*x^8 + 6784*x^7 + 7168*x^6 + 14336/5*x^5 + 3584*x^4 + 8192*x^3 + 8192*x^2 + 4096*x