\(\int (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4)^4 \, dx\) [37]

Optimal result

Integrand size = 32, antiderivative size = 266 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=\frac {\left (5 d^4+256 a e^3\right )^4 x}{1048576 e^4}-\frac {d^2 \left (5 d^4+256 a e^3\right )^3 (d+4 e x)^3}{524288 e^5}+\frac {\left (5 d^4+256 a e^3\right )^2 \left (59 d^4+256 a e^3\right ) (d+4 e x)^5}{5242880 e^5}-\frac {9 d^2 \left (5 d^4+256 a e^3\right ) \left (17 d^4+256 a e^3\right ) (d+4 e x)^7}{3670016 e^5}+\frac {\left (601 d^8+20992 a d^4 e^3+65536 a^2 e^6\right ) (d+4 e x)^9}{6291456 e^5}-\frac {9 d^2 \left (17 d^4+256 a e^3\right ) (d+4 e x)^{11}}{5767168 e^5}+\frac {\left (59 d^4+256 a e^3\right ) (d+4 e x)^{13}}{13631488 e^5}-\frac {d^2 (d+4 e x)^{15}}{2621440 e^5}+\frac {(d+4 e x)^{17}}{71303168 e^5} \] Output:

1/1048576*(256*a*e^3+5*d^4)^4*x/e^4-1/524288*d^2*(256*a*e^3+5*d^4)^3*(4*e* 
x+d)^3/e^5+1/5242880*(256*a*e^3+5*d^4)^2*(256*a*e^3+59*d^4)*(4*e*x+d)^5/e^ 
5-9/3670016*d^2*(256*a*e^3+5*d^4)*(256*a*e^3+17*d^4)*(4*e*x+d)^7/e^5+1/629 
1456*(65536*a^2*e^6+20992*a*d^4*e^3+601*d^8)*(4*e*x+d)^9/e^5-9/5767168*d^2 
*(256*a*e^3+17*d^4)*(4*e*x+d)^11/e^5+1/13631488*(256*a*e^3+59*d^4)*(4*e*x+ 
d)^13/e^5-1/2621440*d^2*(4*e*x+d)^15/e^5+1/71303168*(4*e*x+d)^17/e^5
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.30 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=4096 a^4 e^8 x-1024 a^3 d^3 e^6 x^2+128 a^2 d^6 e^4 x^3+8 a d e^2 \left (-d^8+512 a^2 e^6\right ) x^4+\frac {1}{5} \left (d^{12}-6144 a^2 d^4 e^6+16384 a^3 e^9\right ) x^5-128 a d^3 e^4 \left (-d^4+8 a e^3\right ) x^6-\frac {32}{7} d^2 e^2 \left (d^8-24 a d^4 e^3-768 a^2 e^6\right ) x^7-4 d e^3 \left (d^8+192 a d^4 e^3-1536 a^2 e^6\right ) x^8+\frac {128}{3} e^4 \left (d^8-32 a d^4 e^3+64 a^2 e^6\right ) x^9+\frac {128}{5} d^3 e^5 \left (3 d^4+40 a e^3\right ) x^{10}+\frac {128}{11} d^2 e^6 \left (-13 d^4+384 a e^3\right ) x^{11}-512 d e^7 \left (d^4-8 a e^3\right ) x^{12}+\frac {2048}{13} e^8 \left (-d^4+8 a e^3\right ) x^{13}+1024 d^3 e^9 x^{14}+\frac {8192}{5} d^2 e^{10} x^{15}+1024 d e^{11} x^{16}+\frac {4096 e^{12} x^{17}}{17} \] Input:

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^4,x]
 

Output:

4096*a^4*e^8*x - 1024*a^3*d^3*e^6*x^2 + 128*a^2*d^6*e^4*x^3 + 8*a*d*e^2*(- 
d^8 + 512*a^2*e^6)*x^4 + ((d^12 - 6144*a^2*d^4*e^6 + 16384*a^3*e^9)*x^5)/5 
 - 128*a*d^3*e^4*(-d^4 + 8*a*e^3)*x^6 - (32*d^2*e^2*(d^8 - 24*a*d^4*e^3 - 
768*a^2*e^6)*x^7)/7 - 4*d*e^3*(d^8 + 192*a*d^4*e^3 - 1536*a^2*e^6)*x^8 + ( 
128*e^4*(d^8 - 32*a*d^4*e^3 + 64*a^2*e^6)*x^9)/3 + (128*d^3*e^5*(3*d^4 + 4 
0*a*e^3)*x^10)/5 + (128*d^2*e^6*(-13*d^4 + 384*a*e^3)*x^11)/11 - 512*d*e^7 
*(d^4 - 8*a*e^3)*x^12 + (2048*e^8*(-d^4 + 8*a*e^3)*x^13)/13 + 1024*d^3*e^9 
*x^14 + (8192*d^2*e^10*x^15)/5 + 1024*d*e^11*x^16 + (4096*e^12*x^17)/17
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2458, 1403, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \left (\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^4d\left (\frac {d}{4 e}+x\right )\)

\(\Big \downarrow \) 1403

\(\displaystyle \int \left (\frac {27}{512} d^4 \left (256 a e^3+5 d^4\right )^2 \left (\frac {1}{54} \left (\frac {256 a e^3}{d^4}+5\right )+1\right ) \left (\frac {d}{4 e}+x\right )^4+3456 d^4 e^8 \left (\frac {1}{54} \left (\frac {256 a e^3}{d^4}+5\right )+1\right ) \left (\frac {d}{4 e}+x\right )^{12}+\frac {\left (256 a e^3+5 d^4\right )^4}{1048576 e^4}+81 d^8 e^4 \left (\frac {\left (256 a e^3+5 d^4\right ) \left (256 a e^3+77 d^4\right )}{216 d^8}+1\right ) \left (\frac {d}{4 e}+x\right )^8-864 d^6 e^6 \left (\frac {64 a e^3}{3 d^4}+\frac {17}{12}\right ) \left (\frac {d}{4 e}+x\right )^{10}-\frac {27}{8} d^6 e^2 \left (256 a e^3+5 d^4\right ) \left (\frac {64 a e^3}{3 d^4}+\frac {17}{12}\right ) \left (\frac {d}{4 e}+x\right )^6-\frac {3 d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac {d}{4 e}+x\right )^2}{8192 e^2}-6144 d^2 e^{10} \left (\frac {d}{4 e}+x\right )^{14}+4096 e^{12} \left (\frac {d}{4 e}+x\right )^{16}\right )d\left (\frac {d}{4 e}+x\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{24} e^4 \left (65536 a^2 e^6+20992 a d^4 e^3+601 d^8\right ) \left (\frac {d}{4 e}+x\right )^9+\frac {\left (256 a e^3+5 d^4\right )^2 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^5}{5120}+\frac {64}{13} e^8 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^{13}+\frac {\left (256 a e^3+5 d^4\right )^4 \left (\frac {d}{4 e}+x\right )}{1048576 e^4}-\frac {72}{11} d^2 e^6 \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^{11}-\frac {9}{224} d^2 e^2 \left (256 a e^3+5 d^4\right ) \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^7-\frac {d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac {d}{4 e}+x\right )^3}{8192 e^2}-\frac {2048}{5} d^2 e^{10} \left (\frac {d}{4 e}+x\right )^{15}+\frac {4096}{17} e^{12} \left (\frac {d}{4 e}+x\right )^{17}\)

Input:

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^4,x]
 

Output:

((5*d^4 + 256*a*e^3)^4*(d/(4*e) + x))/(1048576*e^4) - (d^2*(5*d^4 + 256*a* 
e^3)^3*(d/(4*e) + x)^3)/(8192*e^2) + ((5*d^4 + 256*a*e^3)^2*(59*d^4 + 256* 
a*e^3)*(d/(4*e) + x)^5)/5120 - (9*d^2*e^2*(5*d^4 + 256*a*e^3)*(17*d^4 + 25 
6*a*e^3)*(d/(4*e) + x)^7)/224 + (e^4*(601*d^8 + 20992*a*d^4*e^3 + 65536*a^ 
2*e^6)*(d/(4*e) + x)^9)/24 - (72*d^2*e^6*(17*d^4 + 256*a*e^3)*(d/(4*e) + x 
)^11)/11 + (64*e^8*(59*d^4 + 256*a*e^3)*(d/(4*e) + x)^13)/13 - (2048*d^2*e 
^10*(d/(4*e) + x)^15)/5 + (4096*e^12*(d/(4*e) + x)^17)/17
 

Defintions of rubi rules used

rule 1403

Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandInte 
grand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a 
*c, 0] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2458

Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp 
on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x 
- S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp 
on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P 
n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.25

method	result	size
norman	\(4096 a^{4} e^{8} x -1024 a^{3} e^{6} d^{3} x^{2}+128 a^{2} e^{4} d^{6} x^{3}+\left (4096 a^{3} e^{8} d -8 a \,e^{2} d^{9}\right ) x^{4}+\left (\frac {16384}{5} a^{3} e^{9}-\frac {6144}{5} a^{2} e^{6} d^{4}+\frac {1}{5} d^{12}\right ) x^{5}+\left (-1024 a^{2} e^{7} d^{3}+128 d^{7} a \,e^{4}\right ) x^{6}+\left (\frac {24576}{7} a^{2} e^{8} d^{2}+\frac {768}{7} a \,e^{5} d^{6}-\frac {32}{7} d^{10} e^{2}\right ) x^{7}+\left (6144 a^{2} e^{9} d -768 a \,e^{6} d^{5}-4 d^{9} e^{3}\right ) x^{8}+\left (\frac {8192}{3} a^{2} e^{10}-\frac {4096}{3} a \,e^{7} d^{4}+\frac {128}{3} d^{8} e^{4}\right ) x^{9}+\left (1024 a \,e^{8} d^{3}+\frac {384}{5} d^{7} e^{5}\right ) x^{10}+\left (\frac {49152}{11} a \,e^{9} d^{2}-\frac {1664}{11} d^{6} e^{6}\right ) x^{11}+\left (4096 a \,e^{10} d -512 d^{5} e^{7}\right ) x^{12}+\left (\frac {16384}{13} a \,e^{11}-\frac {2048}{13} d^{4} e^{8}\right ) x^{13}+1024 d^{3} e^{9} x^{14}+\frac {8192 d^{2} e^{10} x^{15}}{5}+1024 d \,e^{11} x^{16}+\frac {4096 e^{12} x^{17}}{17}\)	\(332\)
gosper	\(4096 a d \,e^{10} x^{12}-768 a \,d^{5} e^{6} x^{8}+6144 a^{2} d \,e^{9} x^{8}-1024 a^{2} d^{3} e^{7} x^{6}+128 a \,d^{7} e^{4} x^{6}-8 a \,d^{9} e^{2} x^{4}-\frac {4096}{3} x^{9} a \,e^{7} d^{4}+1024 x^{10} a \,e^{8} d^{3}+\frac {49152}{11} x^{11} a \,e^{9} d^{2}+4096 a^{3} d \,e^{8} x^{4}-\frac {6144}{5} x^{5} a^{2} e^{6} d^{4}+\frac {24576}{7} x^{7} a^{2} e^{8} d^{2}+\frac {768}{7} x^{7} a \,e^{5} d^{6}+128 a^{2} e^{4} d^{6} x^{3}-1024 a^{3} e^{6} d^{3} x^{2}+\frac {4096}{17} e^{12} x^{17}+\frac {1}{5} x^{5} d^{12}+1024 d^{3} e^{9} x^{14}+\frac {8192}{5} d^{2} e^{10} x^{15}+1024 d \,e^{11} x^{16}+\frac {16384}{5} x^{5} a^{3} e^{9}-\frac {32}{7} x^{7} d^{10} e^{2}+\frac {8192}{3} x^{9} a^{2} e^{10}+\frac {128}{3} x^{9} d^{8} e^{4}+\frac {384}{5} x^{10} d^{7} e^{5}-\frac {1664}{11} x^{11} d^{6} e^{6}+\frac {16384}{13} x^{13} a \,e^{11}-\frac {2048}{13} x^{13} d^{4} e^{8}-4 d^{9} e^{3} x^{8}-512 d^{5} e^{7} x^{12}+4096 a^{4} e^{8} x\)	\(354\)
risch	\(4096 a d \,e^{10} x^{12}-768 a \,d^{5} e^{6} x^{8}+6144 a^{2} d \,e^{9} x^{8}-1024 a^{2} d^{3} e^{7} x^{6}+128 a \,d^{7} e^{4} x^{6}-8 a \,d^{9} e^{2} x^{4}-\frac {4096}{3} x^{9} a \,e^{7} d^{4}+1024 x^{10} a \,e^{8} d^{3}+\frac {49152}{11} x^{11} a \,e^{9} d^{2}+4096 a^{3} d \,e^{8} x^{4}-\frac {6144}{5} x^{5} a^{2} e^{6} d^{4}+\frac {24576}{7} x^{7} a^{2} e^{8} d^{2}+\frac {768}{7} x^{7} a \,e^{5} d^{6}+128 a^{2} e^{4} d^{6} x^{3}-1024 a^{3} e^{6} d^{3} x^{2}+\frac {4096}{17} e^{12} x^{17}+\frac {1}{5} x^{5} d^{12}+1024 d^{3} e^{9} x^{14}+\frac {8192}{5} d^{2} e^{10} x^{15}+1024 d \,e^{11} x^{16}+\frac {16384}{5} x^{5} a^{3} e^{9}-\frac {32}{7} x^{7} d^{10} e^{2}+\frac {8192}{3} x^{9} a^{2} e^{10}+\frac {128}{3} x^{9} d^{8} e^{4}+\frac {384}{5} x^{10} d^{7} e^{5}-\frac {1664}{11} x^{11} d^{6} e^{6}+\frac {16384}{13} x^{13} a \,e^{11}-\frac {2048}{13} x^{13} d^{4} e^{8}-4 d^{9} e^{3} x^{8}-512 d^{5} e^{7} x^{12}+4096 a^{4} e^{8} x\)	\(354\)
parallelrisch	\(4096 a d \,e^{10} x^{12}-768 a \,d^{5} e^{6} x^{8}+6144 a^{2} d \,e^{9} x^{8}-1024 a^{2} d^{3} e^{7} x^{6}+128 a \,d^{7} e^{4} x^{6}-8 a \,d^{9} e^{2} x^{4}-\frac {4096}{3} x^{9} a \,e^{7} d^{4}+1024 x^{10} a \,e^{8} d^{3}+\frac {49152}{11} x^{11} a \,e^{9} d^{2}+4096 a^{3} d \,e^{8} x^{4}-\frac {6144}{5} x^{5} a^{2} e^{6} d^{4}+\frac {24576}{7} x^{7} a^{2} e^{8} d^{2}+\frac {768}{7} x^{7} a \,e^{5} d^{6}+128 a^{2} e^{4} d^{6} x^{3}-1024 a^{3} e^{6} d^{3} x^{2}+\frac {4096}{17} e^{12} x^{17}+\frac {1}{5} x^{5} d^{12}+1024 d^{3} e^{9} x^{14}+\frac {8192}{5} d^{2} e^{10} x^{15}+1024 d \,e^{11} x^{16}+\frac {16384}{5} x^{5} a^{3} e^{9}-\frac {32}{7} x^{7} d^{10} e^{2}+\frac {8192}{3} x^{9} a^{2} e^{10}+\frac {128}{3} x^{9} d^{8} e^{4}+\frac {384}{5} x^{10} d^{7} e^{5}-\frac {1664}{11} x^{11} d^{6} e^{6}+\frac {16384}{13} x^{13} a \,e^{11}-\frac {2048}{13} x^{13} d^{4} e^{8}-4 d^{9} e^{3} x^{8}-512 d^{5} e^{7} x^{12}+4096 a^{4} e^{8} x\)	\(354\)
orering	\(\frac {x \left (61501440 e^{12} x^{16}[phone] d \,e^{11} x^{15}[phone] d^{2} e^{10} x^{14}[phone] d^{3} e^{9} x^{13}[phone] a \,e^{11} x^{12}-40212480 d^{4} e^{8} x^{12}[phone] a d \,e^{10} x^{11}-130690560 d^{5} e^{7} x^{11}[phone] a \,d^{2} e^{9} x^{10}-38613120 d^{6} e^{6} x^{10}[phone] a \,d^{3} e^{8} x^{9}[phone] d^{7} e^{5} x^{9}[phone] a^{2} e^{10} x^{8}-348508160 a \,d^{4} e^{7} x^{8}[phone] d^{8} e^{4} x^{8}[phone] a^{2} d \,e^{9} x^{7}-196035840 a \,d^{5} e^{6} x^{7}-1021020 d^{9} e^{3} x^{7}[phone] a^{2} d^{2} e^{8} x^{6}[phone] a \,d^{6} e^{5} x^{6}-1166880 d^{10} e^{2} x^{6}-261381120 a^{2} d^{3} e^{7} x^{5}[phone] a \,d^{7} e^{4} x^{5}[phone] a^{3} e^{9} x^{4}-313657344 a^{2} d^{4} e^{6} x^{4}+51051 d^{12} x^{4}[phone] a^{3} d \,e^{8} x^{3}-2042040 a \,d^{9} e^{2} x^{3}[phone] a^{2} e^{4} d^{6} x^{2}-261381120 a^{3} e^{6} d^{3} x [phone] a^{4} e^{8}\right )}{255255}\)	\(354\)
default	\(\frac {4096 e^{12} x^{17}}{17}+1024 d \,e^{11} x^{16}+\frac {8192 d^{2} e^{10} x^{15}}{5}+1024 d^{3} e^{9} x^{14}+\frac {128 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) e^{6} x^{13}}{13}+\frac {\left (16384 a \,e^{10} d +256 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d \,e^{5}-2048 d^{5} e^{7}\right ) x^{12}}{12}+\frac {\left (384 d^{6} e^{6}+32768 a \,e^{9} d^{2}+128 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{2} e^{4}\right ) x^{11}}{11}+\frac {\left (14336 a \,e^{8} d^{3}+256 d^{7} e^{5}-32 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{3} e^{3}\right ) x^{10}}{10}+\frac {\left (8192 a^{2} e^{10}-8192 a \,e^{7} d^{4}+128 d^{8} e^{4}+\left (128 a \,e^{5}-16 d^{4} e^{2}\right )^{2}\right ) x^{9}}{9}+\frac {\left (16384 a^{2} e^{9} d -2048 a \,e^{6} d^{5}-32 d^{9} e^{3}+256 a \,e^{4} d \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{8}}{8}+\frac {\left (24576 a^{2} e^{8} d^{2}+512 a \,e^{5} d^{6}+2 d^{6} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{7}}{7}+\frac {\left (-2048 a^{2} e^{7} d^{3}-32 a \,e^{2} d^{3} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )+256 d^{7} a \,e^{4}\right ) x^{6}}{6}+\frac {\left (128 a^{2} e^{4} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )-4096 a^{2} e^{6} d^{4}+d^{12}\right ) x^{5}}{5}+\frac {\left (16384 a^{3} e^{8} d -32 a \,e^{2} d^{9}\right ) x^{4}}{4}+128 a^{2} e^{4} d^{6} x^{3}-1024 a^{3} e^{6} d^{3} x^{2}+4096 a^{4} e^{8} x\)	\(500\)

Input:

int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^4,x,method=_RETURNVERBOSE)
 

Output:

4096*a^4*e^8*x-1024*a^3*e^6*d^3*x^2+128*a^2*e^4*d^6*x^3+(4096*a^3*d*e^8-8* 
a*d^9*e^2)*x^4+(16384/5*a^3*e^9-6144/5*a^2*e^6*d^4+1/5*d^12)*x^5+(-1024*a^ 
2*d^3*e^7+128*a*d^7*e^4)*x^6+(24576/7*a^2*e^8*d^2+768/7*a*e^5*d^6-32/7*d^1 
0*e^2)*x^7+(6144*a^2*d*e^9-768*a*d^5*e^6-4*d^9*e^3)*x^8+(8192/3*a^2*e^10-4 
096/3*a*e^7*d^4+128/3*d^8*e^4)*x^9+(1024*a*e^8*d^3+384/5*d^7*e^5)*x^10+(49 
152/11*a*e^9*d^2-1664/11*d^6*e^6)*x^11+(4096*a*d*e^10-512*d^5*e^7)*x^12+(1 
6384/13*a*e^11-2048/13*d^4*e^8)*x^13+1024*d^3*e^9*x^14+8192/5*d^2*e^10*x^1 
5+1024*d*e^11*x^16+4096/17*e^12*x^17
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.25 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=\frac {4096}{17} \, e^{12} x^{17} + 1024 \, d e^{11} x^{16} + \frac {8192}{5} \, d^{2} e^{10} x^{15} + 1024 \, d^{3} e^{9} x^{14} + 128 \, a^{2} d^{6} e^{4} x^{3} - 1024 \, a^{3} d^{3} e^{6} x^{2} - \frac {2048}{13} \, {\left (d^{4} e^{8} - 8 \, a e^{11}\right )} x^{13} + 4096 \, a^{4} e^{8} x - 512 \, {\left (d^{5} e^{7} - 8 \, a d e^{10}\right )} x^{12} - \frac {128}{11} \, {\left (13 \, d^{6} e^{6} - 384 \, a d^{2} e^{9}\right )} x^{11} + \frac {128}{5} \, {\left (3 \, d^{7} e^{5} + 40 \, a d^{3} e^{8}\right )} x^{10} + \frac {128}{3} \, {\left (d^{8} e^{4} - 32 \, a d^{4} e^{7} + 64 \, a^{2} e^{10}\right )} x^{9} - 4 \, {\left (d^{9} e^{3} + 192 \, a d^{5} e^{6} - 1536 \, a^{2} d e^{9}\right )} x^{8} - \frac {32}{7} \, {\left (d^{10} e^{2} - 24 \, a d^{6} e^{5} - 768 \, a^{2} d^{2} e^{8}\right )} x^{7} + 128 \, {\left (a d^{7} e^{4} - 8 \, a^{2} d^{3} e^{7}\right )} x^{6} + \frac {1}{5} \, {\left (d^{12} - 6144 \, a^{2} d^{4} e^{6} + 16384 \, a^{3} e^{9}\right )} x^{5} - 8 \, {\left (a d^{9} e^{2} - 512 \, a^{3} d e^{8}\right )} x^{4} \] Input:

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^4,x, algorithm="fricas")
 

Output:

4096/17*e^12*x^17 + 1024*d*e^11*x^16 + 8192/5*d^2*e^10*x^15 + 1024*d^3*e^9 
*x^14 + 128*a^2*d^6*e^4*x^3 - 1024*a^3*d^3*e^6*x^2 - 2048/13*(d^4*e^8 - 8* 
a*e^11)*x^13 + 4096*a^4*e^8*x - 512*(d^5*e^7 - 8*a*d*e^10)*x^12 - 128/11*( 
13*d^6*e^6 - 384*a*d^2*e^9)*x^11 + 128/5*(3*d^7*e^5 + 40*a*d^3*e^8)*x^10 + 
 128/3*(d^8*e^4 - 32*a*d^4*e^7 + 64*a^2*e^10)*x^9 - 4*(d^9*e^3 + 192*a*d^5 
*e^6 - 1536*a^2*d*e^9)*x^8 - 32/7*(d^10*e^2 - 24*a*d^6*e^5 - 768*a^2*d^2*e 
^8)*x^7 + 128*(a*d^7*e^4 - 8*a^2*d^3*e^7)*x^6 + 1/5*(d^12 - 6144*a^2*d^4*e 
^6 + 16384*a^3*e^9)*x^5 - 8*(a*d^9*e^2 - 512*a^3*d*e^8)*x^4
 

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.38 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=4096 a^{4} e^{8} x - 1024 a^{3} d^{3} e^{6} x^{2} + 128 a^{2} d^{6} e^{4} x^{3} + 1024 d^{3} e^{9} x^{14} + \frac {8192 d^{2} e^{10} x^{15}}{5} + 1024 d e^{11} x^{16} + \frac {4096 e^{12} x^{17}}{17} + x^{13} \cdot \left (\frac {16384 a e^{11}}{13} - \frac {2048 d^{4} e^{8}}{13}\right ) + x^{12} \cdot \left (4096 a d e^{10} - 512 d^{5} e^{7}\right ) + x^{11} \cdot \left (\frac {49152 a d^{2} e^{9}}{11} - \frac {1664 d^{6} e^{6}}{11}\right ) + x^{10} \cdot \left (1024 a d^{3} e^{8} + \frac {384 d^{7} e^{5}}{5}\right ) + x^{9} \cdot \left (\frac {8192 a^{2} e^{10}}{3} - \frac {4096 a d^{4} e^{7}}{3} + \frac {128 d^{8} e^{4}}{3}\right ) + x^{8} \cdot \left (6144 a^{2} d e^{9} - 768 a d^{5} e^{6} - 4 d^{9} e^{3}\right ) + x^{7} \cdot \left (\frac {24576 a^{2} d^{2} e^{8}}{7} + \frac {768 a d^{6} e^{5}}{7} - \frac {32 d^{10} e^{2}}{7}\right ) + x^{6} \left (- 1024 a^{2} d^{3} e^{7} + 128 a d^{7} e^{4}\right ) + x^{5} \cdot \left (\frac {16384 a^{3} e^{9}}{5} - \frac {6144 a^{2} d^{4} e^{6}}{5} + \frac {d^{12}}{5}\right ) + x^{4} \cdot \left (4096 a^{3} d e^{8} - 8 a d^{9} e^{2}\right ) \] Input:

integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**4,x)
 

Output:

4096*a**4*e**8*x - 1024*a**3*d**3*e**6*x**2 + 128*a**2*d**6*e**4*x**3 + 10 
24*d**3*e**9*x**14 + 8192*d**2*e**10*x**15/5 + 1024*d*e**11*x**16 + 4096*e 
**12*x**17/17 + x**13*(16384*a*e**11/13 - 2048*d**4*e**8/13) + x**12*(4096 
*a*d*e**10 - 512*d**5*e**7) + x**11*(49152*a*d**2*e**9/11 - 1664*d**6*e**6 
/11) + x**10*(1024*a*d**3*e**8 + 384*d**7*e**5/5) + x**9*(8192*a**2*e**10/ 
3 - 4096*a*d**4*e**7/3 + 128*d**8*e**4/3) + x**8*(6144*a**2*d*e**9 - 768*a 
*d**5*e**6 - 4*d**9*e**3) + x**7*(24576*a**2*d**2*e**8/7 + 768*a*d**6*e**5 
/7 - 32*d**10*e**2/7) + x**6*(-1024*a**2*d**3*e**7 + 128*a*d**7*e**4) + x* 
*5*(16384*a**3*e**9/5 - 6144*a**2*d**4*e**6/5 + d**12/5) + x**4*(4096*a**3 
*d*e**8 - 8*a*d**9*e**2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.44 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=\frac {4096}{17} \, e^{12} x^{17} + 1024 \, d e^{11} x^{16} + \frac {8192}{5} \, d^{2} e^{10} x^{15} + \frac {8192}{7} \, d^{3} e^{9} x^{14} + \frac {4096}{13} \, d^{4} e^{8} x^{13} + \frac {1}{5} \, d^{12} x^{5} + 4096 \, a^{4} e^{8} x - \frac {4}{7} \, {\left (7 \, e^{3} x^{8} + 8 \, d e^{2} x^{7}\right )} d^{9} + \frac {1024}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{3} e^{6} + \frac {128}{165} \, {\left (45 \, e^{6} x^{11} + 99 \, d e^{5} x^{10} + 55 \, d^{2} e^{4} x^{9}\right )} d^{6} + \frac {128}{105} \, {\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\right )} a^{2} e^{4} - \frac {512}{1001} \, {\left (286 \, e^{9} x^{14} + 924 \, d e^{8} x^{13} + 1001 \, d^{2} e^{7} x^{12} + 364 \, d^{3} e^{6} x^{11}\right )} d^{3} + \frac {8}{15015} \, {\left (2365440 \, e^{9} x^{13} + 7687680 \, d e^{8} x^{12} + 8386560 \, d^{2} e^{7} x^{11} + 3075072 \, d^{3} e^{6} x^{10} - 15015 \, d^{9} x^{4} + 34320 \, {\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} - 32032 \, {\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3}\right )} a e^{2} \] Input:

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^4,x, algorithm="maxima")
 

Output:

4096/17*e^12*x^17 + 1024*d*e^11*x^16 + 8192/5*d^2*e^10*x^15 + 8192/7*d^3*e 
^9*x^14 + 4096/13*d^4*e^8*x^13 + 1/5*d^12*x^5 + 4096*a^4*e^8*x - 4/7*(7*e^ 
3*x^8 + 8*d*e^2*x^7)*d^9 + 1024/5*(16*e^3*x^5 + 20*d*e^2*x^4 - 5*d^3*x^2)* 
a^3*e^6 + 128/165*(45*e^6*x^11 + 99*d*e^5*x^10 + 55*d^2*e^4*x^9)*d^6 + 128 
/105*(2240*e^6*x^9 + 5040*d*e^5*x^8 + 2880*d^2*e^4*x^7 + 105*d^6*x^3 - 168 
*(5*e^3*x^6 + 6*d*e^2*x^5)*d^3)*a^2*e^4 - 512/1001*(286*e^9*x^14 + 924*d*e 
^8*x^13 + 1001*d^2*e^7*x^12 + 364*d^3*e^6*x^11)*d^3 + 8/15015*(2365440*e^9 
*x^13 + 7687680*d*e^8*x^12 + 8386560*d^2*e^7*x^11 + 3075072*d^3*e^6*x^10 - 
 15015*d^9*x^4 + 34320*(6*e^3*x^7 + 7*d*e^2*x^6)*d^6 - 32032*(36*e^6*x^10 
+ 80*d*e^5*x^9 + 45*d^2*e^4*x^8)*d^3)*a*e^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.33 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=\frac {4096}{17} \, e^{12} x^{17} + 1024 \, d e^{11} x^{16} + \frac {8192}{5} \, d^{2} e^{10} x^{15} + 1024 \, d^{3} e^{9} x^{14} - \frac {2048}{13} \, d^{4} e^{8} x^{13} + \frac {16384}{13} \, a e^{11} x^{13} - 512 \, d^{5} e^{7} x^{12} + 4096 \, a d e^{10} x^{12} - \frac {1664}{11} \, d^{6} e^{6} x^{11} + \frac {49152}{11} \, a d^{2} e^{9} x^{11} + \frac {384}{5} \, d^{7} e^{5} x^{10} + 1024 \, a d^{3} e^{8} x^{10} + \frac {128}{3} \, d^{8} e^{4} x^{9} - \frac {4096}{3} \, a d^{4} e^{7} x^{9} + \frac {8192}{3} \, a^{2} e^{10} x^{9} - 4 \, d^{9} e^{3} x^{8} - 768 \, a d^{5} e^{6} x^{8} + 6144 \, a^{2} d e^{9} x^{8} - \frac {32}{7} \, d^{10} e^{2} x^{7} + \frac {768}{7} \, a d^{6} e^{5} x^{7} + \frac {24576}{7} \, a^{2} d^{2} e^{8} x^{7} + 128 \, a d^{7} e^{4} x^{6} - 1024 \, a^{2} d^{3} e^{7} x^{6} + \frac {1}{5} \, d^{12} x^{5} - \frac {6144}{5} \, a^{2} d^{4} e^{6} x^{5} + \frac {16384}{5} \, a^{3} e^{9} x^{5} - 8 \, a d^{9} e^{2} x^{4} + 4096 \, a^{3} d e^{8} x^{4} + 128 \, a^{2} d^{6} e^{4} x^{3} - 1024 \, a^{3} d^{3} e^{6} x^{2} + 4096 \, a^{4} e^{8} x \] Input:

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^4,x, algorithm="giac")
 

Output:

4096/17*e^12*x^17 + 1024*d*e^11*x^16 + 8192/5*d^2*e^10*x^15 + 1024*d^3*e^9 
*x^14 - 2048/13*d^4*e^8*x^13 + 16384/13*a*e^11*x^13 - 512*d^5*e^7*x^12 + 4 
096*a*d*e^10*x^12 - 1664/11*d^6*e^6*x^11 + 49152/11*a*d^2*e^9*x^11 + 384/5 
*d^7*e^5*x^10 + 1024*a*d^3*e^8*x^10 + 128/3*d^8*e^4*x^9 - 4096/3*a*d^4*e^7 
*x^9 + 8192/3*a^2*e^10*x^9 - 4*d^9*e^3*x^8 - 768*a*d^5*e^6*x^8 + 6144*a^2* 
d*e^9*x^8 - 32/7*d^10*e^2*x^7 + 768/7*a*d^6*e^5*x^7 + 24576/7*a^2*d^2*e^8* 
x^7 + 128*a*d^7*e^4*x^6 - 1024*a^2*d^3*e^7*x^6 + 1/5*d^12*x^5 - 6144/5*a^2 
*d^4*e^6*x^5 + 16384/5*a^3*e^9*x^5 - 8*a*d^9*e^2*x^4 + 4096*a^3*d*e^8*x^4 
+ 128*a^2*d^6*e^4*x^3 - 1024*a^3*d^3*e^6*x^2 + 4096*a^4*e^8*x
 

Mupad [B] (verification not implemented)

Time = 21.80 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.24 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=x^5\,\left (\frac {16384\,a^3\,e^9}{5}-\frac {6144\,a^2\,d^4\,e^6}{5}+\frac {d^{12}}{5}\right )+x^{10}\,\left (\frac {384\,d^7\,e^5}{5}+1024\,a\,d^3\,e^8\right )-x^{11}\,\left (\frac {1664\,d^6\,e^6}{11}-\frac {49152\,a\,d^2\,e^9}{11}\right )+\frac {4096\,e^{12}\,x^{17}}{17}+\frac {2048\,e^8\,x^{13}\,\left (8\,a\,e^3-d^4\right )}{13}+\frac {128\,e^4\,x^9\,\left (64\,a^2\,e^6-32\,a\,d^4\,e^3+d^8\right )}{3}+4096\,a^4\,e^8\,x+1024\,d\,e^{11}\,x^{16}+1024\,d^3\,e^9\,x^{14}+\frac {8192\,d^2\,e^{10}\,x^{15}}{5}+512\,d\,e^7\,x^{12}\,\left (8\,a\,e^3-d^4\right )+\frac {32\,d^2\,e^2\,x^7\,\left (768\,a^2\,e^6+24\,a\,d^4\,e^3-d^8\right )}{7}-1024\,a^3\,d^3\,e^6\,x^2+128\,a^2\,d^6\,e^4\,x^3-4\,d\,e^3\,x^8\,\left (-1536\,a^2\,e^6+192\,a\,d^4\,e^3+d^8\right )-128\,a\,d^3\,e^4\,x^6\,\left (8\,a\,e^3-d^4\right )-8\,a\,d\,e^2\,x^4\,\left (d^8-512\,a^2\,e^6\right ) \] Input:

int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^4,x)
 

Output:

x^5*(d^12/5 + (16384*a^3*e^9)/5 - (6144*a^2*d^4*e^6)/5) + x^10*((384*d^7*e 
^5)/5 + 1024*a*d^3*e^8) - x^11*((1664*d^6*e^6)/11 - (49152*a*d^2*e^9)/11) 
+ (4096*e^12*x^17)/17 + (2048*e^8*x^13*(8*a*e^3 - d^4))/13 + (128*e^4*x^9* 
(d^8 + 64*a^2*e^6 - 32*a*d^4*e^3))/3 + 4096*a^4*e^8*x + 1024*d*e^11*x^16 + 
 1024*d^3*e^9*x^14 + (8192*d^2*e^10*x^15)/5 + 512*d*e^7*x^12*(8*a*e^3 - d^ 
4) + (32*d^2*e^2*x^7*(768*a^2*e^6 - d^8 + 24*a*d^4*e^3))/7 - 1024*a^3*d^3* 
e^6*x^2 + 128*a^2*d^6*e^4*x^3 - 4*d*e^3*x^8*(d^8 - 1536*a^2*e^6 + 192*a*d^ 
4*e^3) - 128*a*d^3*e^4*x^6*(8*a*e^3 - d^4) - 8*a*d*e^2*x^4*(d^8 - 512*a^2* 
e^6)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.33 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx=\frac {x \left (61501440 e^{12} x^{16}[phone] d \,e^{11} x^{15}[phone] d^{2} e^{10} x^{14}[phone] d^{3} e^{9} x^{13}[phone] a \,e^{11} x^{12}-40212480 d^{4} e^{8} x^{12}[phone] a d \,e^{10} x^{11}-130690560 d^{5} e^{7} x^{11}[phone] a \,d^{2} e^{9} x^{10}-38613120 d^{6} e^{6} x^{10}[phone] a \,d^{3} e^{8} x^{9}[phone] d^{7} e^{5} x^{9}[phone] a^{2} e^{10} x^{8}-348508160 a \,d^{4} e^{7} x^{8}[phone] d^{8} e^{4} x^{8}[phone] a^{2} d \,e^{9} x^{7}-196035840 a \,d^{5} e^{6} x^{7}-1021020 d^{9} e^{3} x^{7}[phone] a^{2} d^{2} e^{8} x^{6}[phone] a \,d^{6} e^{5} x^{6}-1166880 d^{10} e^{2} x^{6}-261381120 a^{2} d^{3} e^{7} x^{5}[phone] a \,d^{7} e^{4} x^{5}[phone] a^{3} e^{9} x^{4}-313657344 a^{2} d^{4} e^{6} x^{4}+51051 d^{12} x^{4}[phone] a^{3} d \,e^{8} x^{3}-2042040 a \,d^{9} e^{2} x^{3}[phone] a^{2} d^{6} e^{4} x^{2}-261381120 a^{3} d^{3} e^{6} x [phone] a^{4} e^{8}\right )}{255255} \] Input:

int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^4,x)
 

Output:

(x*(1045524480*a**4*e**8 - 261381120*a**3*d**3*e**6*x + 1045524480*a**3*d* 
e**8*x**3 + 836419584*a**3*e**9*x**4 + 32672640*a**2*d**6*e**4*x**2 - 3136 
57344*a**2*d**4*e**6*x**4 - 261381120*a**2*d**3*e**7*x**5 + 896163840*a**2 
*d**2*e**8*x**6 + 1568286720*a**2*d*e**9*x**7 + 697016320*a**2*e**10*x**8 
- 2042040*a*d**9*e**2*x**3 + 32672640*a*d**7*e**4*x**5 + 28005120*a*d**6*e 
**5*x**6 - 196035840*a*d**5*e**6*x**7 - 348508160*a*d**4*e**7*x**8 + 26138 
1120*a*d**3*e**8*x**9 + 1140572160*a*d**2*e**9*x**10 + 1045524480*a*d*e**1 
0*x**11 + 321699840*a*e**11*x**12 + 51051*d**12*x**4 - 1166880*d**10*e**2* 
x**6 - 1021020*d**9*e**3*x**7 + 10890880*d**8*e**4*x**8 + 19603584*d**7*e* 
*5*x**9 - 38613120*d**6*e**6*x**10 - 130690560*d**5*e**7*x**11 - 40212480* 
d**4*e**8*x**12 + 261381120*d**3*e**9*x**13 + 418209792*d**2*e**10*x**14 + 
 261381120*d*e**11*x**15 + 61501440*e**12*x**16))/255255