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

Optimal. Leaf size=104 \[ \frac{4096 x^{17}}{17}-1920 x^{16}+\frac{102784 x^{15}}{15}-\frac{75504 x^{14}}{7}-\frac{12095 x^{13}}{13}+31128 x^{12}-\frac{331040 x^{11}}{11}-\frac{169584 x^{10}}{5}+\frac{641152 x^9}{9}+36384 x^8-\frac{566912 x^7}{7}-30720 x^6+\frac{538624 x^5}{5}+139776 x^4+\frac{237568 x^3}{3}+24576 x^2+4096 x \]

[Out]

4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720*x^6 - (566912*x^7)/7 + 36384*x^8 + (
641152*x^9)/9 - (169584*x^10)/5 - (331040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x
^15)/15 - 1920*x^16 + (4096*x^17)/17

________________________________________________________________________________________

Rubi [A]  time = 0.0337367, antiderivative size = 104, 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} \[ \frac{4096 x^{17}}{17}-1920 x^{16}+\frac{102784 x^{15}}{15}-\frac{75504 x^{14}}{7}-\frac{12095 x^{13}}{13}+31128 x^{12}-\frac{331040 x^{11}}{11}-\frac{169584 x^{10}}{5}+\frac{641152 x^9}{9}+36384 x^8-\frac{566912 x^7}{7}-30720 x^6+\frac{538624 x^5}{5}+139776 x^4+\frac{237568 x^3}{3}+24576 x^2+4096 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720*x^6 - (566912*x^7)/7 + 36384*x^8 + (
641152*x^9)/9 - (169584*x^10)/5 - (331040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x
^15)/15 - 1920*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+24 x+8 x^2-15 x^3+8 x^4\right )^4 \, dx &=\int \left (4096+49152 x+237568 x^2+559104 x^3+538624 x^4-184320 x^5-566912 x^6+291072 x^7+641152 x^8-339168 x^9-331040 x^{10}+373536 x^{11}-12095 x^{12}-151008 x^{13}+102784 x^{14}-30720 x^{15}+4096 x^{16}\right ) \, dx\\ &=4096 x+24576 x^2+\frac{237568 x^3}{3}+139776 x^4+\frac{538624 x^5}{5}-30720 x^6-\frac{566912 x^7}{7}+36384 x^8+\frac{641152 x^9}{9}-\frac{169584 x^{10}}{5}-\frac{331040 x^{11}}{11}+31128 x^{12}-\frac{12095 x^{13}}{13}-\frac{75504 x^{14}}{7}+\frac{102784 x^{15}}{15}-1920 x^{16}+\frac{4096 x^{17}}{17}\\ \end{align*}

Mathematica [A]  time = 0.0016699, size = 104, normalized size = 1. \[ \frac{4096 x^{17}}{17}-1920 x^{16}+\frac{102784 x^{15}}{15}-\frac{75504 x^{14}}{7}-\frac{12095 x^{13}}{13}+31128 x^{12}-\frac{331040 x^{11}}{11}-\frac{169584 x^{10}}{5}+\frac{641152 x^9}{9}+36384 x^8-\frac{566912 x^7}{7}-30720 x^6+\frac{538624 x^5}{5}+139776 x^4+\frac{237568 x^3}{3}+24576 x^2+4096 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

4096*x + 24576*x^2 + (237568*x^3)/3 + 139776*x^4 + (538624*x^5)/5 - 30720*x^6 - (566912*x^7)/7 + 36384*x^8 + (
641152*x^9)/9 - (169584*x^10)/5 - (331040*x^11)/11 + 31128*x^12 - (12095*x^13)/13 - (75504*x^14)/7 + (102784*x
^15)/15 - 1920*x^16 + (4096*x^17)/17

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 85, normalized size = 0.8 \begin{align*} 4096\,x+24576\,{x}^{2}+{\frac{237568\,{x}^{3}}{3}}+139776\,{x}^{4}+{\frac{538624\,{x}^{5}}{5}}-30720\,{x}^{6}-{\frac{566912\,{x}^{7}}{7}}+36384\,{x}^{8}+{\frac{641152\,{x}^{9}}{9}}-{\frac{169584\,{x}^{10}}{5}}-{\frac{331040\,{x}^{11}}{11}}+31128\,{x}^{12}-{\frac{12095\,{x}^{13}}{13}}-{\frac{75504\,{x}^{14}}{7}}+{\frac{102784\,{x}^{15}}{15}}-1920\,{x}^{16}+{\frac{4096\,{x}^{17}}{17}} \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)^4,x)

[Out]

4096*x+24576*x^2+237568/3*x^3+139776*x^4+538624/5*x^5-30720*x^6-566912/7*x^7+36384*x^8+641152/9*x^9-169584/5*x
^10-331040/11*x^11+31128*x^12-12095/13*x^13-75504/7*x^14+102784/15*x^15-1920*x^16+4096/17*x^17

________________________________________________________________________________________

Maxima [A]  time = 1.20434, size = 113, normalized size = 1.09 \begin{align*} \frac{4096}{17} \, x^{17} - 1920 \, x^{16} + \frac{102784}{15} \, x^{15} - \frac{75504}{7} \, x^{14} - \frac{12095}{13} \, x^{13} + 31128 \, x^{12} - \frac{331040}{11} \, x^{11} - \frac{169584}{5} \, x^{10} + \frac{641152}{9} \, x^{9} + 36384 \, x^{8} - \frac{566912}{7} \, x^{7} - 30720 \, x^{6} + \frac{538624}{5} \, x^{5} + 139776 \, x^{4} + \frac{237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, 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)^4,x, algorithm="maxima")

[Out]

4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 16958
4/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 24
576*x^2 + 4096*x

________________________________________________________________________________________

Fricas [A]  time = 1.23255, size = 321, normalized size = 3.09 \begin{align*} \frac{4096}{17} x^{17} - 1920 x^{16} + \frac{102784}{15} x^{15} - \frac{75504}{7} x^{14} - \frac{12095}{13} x^{13} + 31128 x^{12} - \frac{331040}{11} x^{11} - \frac{169584}{5} x^{10} + \frac{641152}{9} x^{9} + 36384 x^{8} - \frac{566912}{7} x^{7} - 30720 x^{6} + \frac{538624}{5} x^{5} + 139776 x^{4} + \frac{237568}{3} x^{3} + 24576 x^{2} + 4096 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)^4,x, algorithm="fricas")

[Out]

4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 16958
4/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 24
576*x^2 + 4096*x

________________________________________________________________________________________

Sympy [A]  time = 0.077191, size = 100, normalized size = 0.96 \begin{align*} \frac{4096 x^{17}}{17} - 1920 x^{16} + \frac{102784 x^{15}}{15} - \frac{75504 x^{14}}{7} - \frac{12095 x^{13}}{13} + 31128 x^{12} - \frac{331040 x^{11}}{11} - \frac{169584 x^{10}}{5} + \frac{641152 x^{9}}{9} + 36384 x^{8} - \frac{566912 x^{7}}{7} - 30720 x^{6} + \frac{538624 x^{5}}{5} + 139776 x^{4} + \frac{237568 x^{3}}{3} + 24576 x^{2} + 4096 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)**4,x)

[Out]

4096*x**17/17 - 1920*x**16 + 102784*x**15/15 - 75504*x**14/7 - 12095*x**13/13 + 31128*x**12 - 331040*x**11/11
- 169584*x**10/5 + 641152*x**9/9 + 36384*x**8 - 566912*x**7/7 - 30720*x**6 + 538624*x**5/5 + 139776*x**4 + 237
568*x**3/3 + 24576*x**2 + 4096*x

________________________________________________________________________________________

Giac [A]  time = 1.1173, size = 113, normalized size = 1.09 \begin{align*} \frac{4096}{17} \, x^{17} - 1920 \, x^{16} + \frac{102784}{15} \, x^{15} - \frac{75504}{7} \, x^{14} - \frac{12095}{13} \, x^{13} + 31128 \, x^{12} - \frac{331040}{11} \, x^{11} - \frac{169584}{5} \, x^{10} + \frac{641152}{9} \, x^{9} + 36384 \, x^{8} - \frac{566912}{7} \, x^{7} - 30720 \, x^{6} + \frac{538624}{5} \, x^{5} + 139776 \, x^{4} + \frac{237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, 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)^4,x, algorithm="giac")

[Out]

4096/17*x^17 - 1920*x^16 + 102784/15*x^15 - 75504/7*x^14 - 12095/13*x^13 + 31128*x^12 - 331040/11*x^11 - 16958
4/5*x^10 + 641152/9*x^9 + 36384*x^8 - 566912/7*x^7 - 30720*x^6 + 538624/5*x^5 + 139776*x^4 + 237568/3*x^3 + 24
576*x^2 + 4096*x