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+237568/3*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

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Rubi [A]  time = 0.03, antiderivative size = 104, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 104, normalized size = 1.00 \[ \frac {4096 x^{17}}{17}-1920 x^{16}+\frac {102784 x^{15}}{15}-\frac {75504 x^{14}}{7}-\frac {12095 x^{13}}{13}+31128 x^{12}-\frac {331040 x^{11}}{11}-\frac {169584 x^{10}}{5}+\frac {641152 x^9}{9}+36384 x^8-\frac {566912 x^7}{7}-30720 x^6+\frac {538624 x^5}{5}+139776 x^4+\frac {237568 x^3}{3}+24576 x^2+4096 x \]

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

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fricas [A]  time = 0.37, size = 84, normalized size = 0.81 \[ \frac {4096}{17} x^{17} - 1920 x^{16} + \frac {102784}{15} x^{15} - \frac {75504}{7} x^{14} - \frac {12095}{13} x^{13} + 31128 x^{12} - \frac {331040}{11} x^{11} - \frac {169584}{5} x^{10} + \frac {641152}{9} x^{9} + 36384 x^{8} - \frac {566912}{7} x^{7} - 30720 x^{6} + \frac {538624}{5} x^{5} + 139776 x^{4} + \frac {237568}{3} x^{3} + 24576 x^{2} + 4096 x \]

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

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giac [A]  time = 0.36, size = 84, normalized size = 0.81 \[ \frac {4096}{17} \, x^{17} - 1920 \, x^{16} + \frac {102784}{15} \, x^{15} - \frac {75504}{7} \, x^{14} - \frac {12095}{13} \, x^{13} + 31128 \, x^{12} - \frac {331040}{11} \, x^{11} - \frac {169584}{5} \, x^{10} + \frac {641152}{9} \, x^{9} + 36384 \, x^{8} - \frac {566912}{7} \, x^{7} - 30720 \, x^{6} + \frac {538624}{5} \, x^{5} + 139776 \, x^{4} + \frac {237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, x \]

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

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maple [A]  time = 0.00, size = 85, normalized size = 0.82 \[ \frac {4096}{17} x^{17}-1920 x^{16}+\frac {102784}{15} x^{15}-\frac {75504}{7} x^{14}-\frac {12095}{13} x^{13}+31128 x^{12}-\frac {331040}{11} x^{11}-\frac {169584}{5} x^{10}+\frac {641152}{9} x^{9}+36384 x^{8}-\frac {566912}{7} x^{7}-30720 x^{6}+\frac {538624}{5} x^{5}+139776 x^{4}+\frac {237568}{3} x^{3}+24576 x^{2}+4096 x \]

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

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maxima [A]  time = 0.66, size = 84, normalized size = 0.81 \[ \frac {4096}{17} \, x^{17} - 1920 \, x^{16} + \frac {102784}{15} \, x^{15} - \frac {75504}{7} \, x^{14} - \frac {12095}{13} \, x^{13} + 31128 \, x^{12} - \frac {331040}{11} \, x^{11} - \frac {169584}{5} \, x^{10} + \frac {641152}{9} \, x^{9} + 36384 \, x^{8} - \frac {566912}{7} \, x^{7} - 30720 \, x^{6} + \frac {538624}{5} \, x^{5} + 139776 \, x^{4} + \frac {237568}{3} \, x^{3} + 24576 \, x^{2} + 4096 \, x \]

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

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mupad [B]  time = 2.23, size = 84, normalized size = 0.81 \[ \frac {4096\,x^{17}}{17}-1920\,x^{16}+\frac {102784\,x^{15}}{15}-\frac {75504\,x^{14}}{7}-\frac {12095\,x^{13}}{13}+31128\,x^{12}-\frac {331040\,x^{11}}{11}-\frac {169584\,x^{10}}{5}+\frac {641152\,x^9}{9}+36384\,x^8-\frac {566912\,x^7}{7}-30720\,x^6+\frac {538624\,x^5}{5}+139776\,x^4+\frac {237568\,x^3}{3}+24576\,x^2+4096\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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sympy [A]  time = 0.08, size = 100, normalized size = 0.96 \[ \frac {4096 x^{17}}{17} - 1920 x^{16} + \frac {102784 x^{15}}{15} - \frac {75504 x^{14}}{7} - \frac {12095 x^{13}}{13} + 31128 x^{12} - \frac {331040 x^{11}}{11} - \frac {169584 x^{10}}{5} + \frac {641152 x^{9}}{9} + 36384 x^{8} - \frac {566912 x^{7}}{7} - 30720 x^{6} + \frac {538624 x^{5}}{5} + 139776 x^{4} + \frac {237568 x^{3}}{3} + 24576 x^{2} + 4096 x \]

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

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