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

Optimal. Leaf size=295 \[ \frac{1}{24} e^4 \left (65536 a^2 e^6+20992 a d^4 e^3+601 d^8\right ) \left (\frac{d}{4 e}+x\right )^9+\frac{64}{13} e^8 \left (256 a e^3+59 d^4\right ) \left (\frac{d}{4 e}+x\right )^{13}-\frac{72}{11} d^2 e^6 \left (256 a e^3+17 d^4\right ) \left (\frac{d}{4 e}+x\right )^{11}-\frac{9}{224} d^2 e^2 \left (256 a e^3+5 d^4\right ) \left (256 a e^3+17 d^4\right ) \left (\frac{d}{4 e}+x\right )^7+\frac{\left (256 a e^3+5 d^4\right )^2 \left (256 a e^3+59 d^4\right ) \left (\frac{d}{4 e}+x\right )^5}{5120}-\frac{d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac{d}{4 e}+x\right )^3}{8192 e^2}+\frac{x \left (256 a e^3+5 d^4\right )^4}{1048576 e^4}-\frac{2048}{5} d^2 e^{10} \left (\frac{d}{4 e}+x\right )^{15}+\frac{4096}{17} e^{12} \left (\frac{d}{4 e}+x\right )^{17} \]

[Out]

((5*d^4 + 256*a*e^3)^4*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 + 2
56*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 + 256*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

________________________________________________________________________________________

Rubi [A]  time = 0.532395, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1106, 1090} \[ \frac{1}{24} e^4 \left (65536 a^2 e^6+20992 a d^4 e^3+601 d^8\right ) \left (\frac{d}{4 e}+x\right )^9+\frac{64}{13} e^8 \left (256 a e^3+59 d^4\right ) \left (\frac{d}{4 e}+x\right )^{13}-\frac{72}{11} d^2 e^6 \left (256 a e^3+17 d^4\right ) \left (\frac{d}{4 e}+x\right )^{11}-\frac{9}{224} d^2 e^2 \left (256 a e^3+5 d^4\right ) \left (256 a e^3+17 d^4\right ) \left (\frac{d}{4 e}+x\right )^7+\frac{\left (256 a e^3+5 d^4\right )^2 \left (256 a e^3+59 d^4\right ) \left (\frac{d}{4 e}+x\right )^5}{5120}-\frac{d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac{d}{4 e}+x\right )^3}{8192 e^2}+\frac{x \left (256 a e^3+5 d^4\right )^4}{1048576 e^4}-\frac{2048}{5} d^2 e^{10} \left (\frac{d}{4 e}+x\right )^{15}+\frac{4096}{17} e^{12} \left (\frac{d}{4 e}+x\right )^{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*d^4 + 256*a*e^3)^4*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 + 2
56*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 + 256*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

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1090

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(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]

Rubi steps

\begin{align*} \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx &=\operatorname{Subst}\left (\int \left (\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4\right )^4 \, dx,x,\frac{d}{4 e}+x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{\left (5 d^4+256 a e^3\right )^4}{1048576 e^4}-\frac{3 d^2 \left (5 d^4+256 a e^3\right )^3 x^2}{8192 e^2}+\frac{27}{512} d^4 \left (5 d^4+256 a e^3\right )^2 \left (1+\frac{1}{54} \left (5+\frac{256 a e^3}{d^4}\right )\right ) x^4-\frac{27}{8} d^6 e^2 \left (5 d^4+256 a e^3\right ) \left (\frac{17}{12}+\frac{64 a e^3}{3 d^4}\right ) x^6+81 d^8 e^4 \left (1+\frac{\left (5 d^4+256 a e^3\right ) \left (77 d^4+256 a e^3\right )}{216 d^8}\right ) x^8-864 d^6 e^6 \left (\frac{17}{12}+\frac{64 a e^3}{3 d^4}\right ) x^{10}+3456 d^4 e^8 \left (1+\frac{1}{54} \left (5+\frac{256 a e^3}{d^4}\right )\right ) x^{12}-6144 d^2 e^{10} x^{14}+4096 e^{12} x^{16}\right ) \, dx,x,\frac{d}{4 e}+x\right )\\ &=\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 \left (\frac{d}{4 e}+x\right )^3}{8192 e^2}+\frac{\left (5 d^4+256 a e^3\right )^2 \left (59 d^4+256 a e^3\right ) \left (\frac{d}{4 e}+x\right )^5}{5120}-\frac{9}{224} d^2 e^2 \left (5 d^4+256 a e^3\right ) \left (17 d^4+256 a e^3\right ) \left (\frac{d}{4 e}+x\right )^7+\frac{1}{24} e^4 \left (601 d^8+20992 a d^4 e^3+65536 a^2 e^6\right ) \left (\frac{d}{4 e}+x\right )^9-\frac{72}{11} d^2 e^6 \left (17 d^4+256 a e^3\right ) \left (\frac{d}{4 e}+x\right )^{11}+\frac{64}{13} e^8 \left (59 d^4+256 a e^3\right ) \left (\frac{d}{4 e}+x\right )^{13}-\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}\\ \end{align*}

Mathematica [A]  time = 0.0489849, size = 345, normalized size = 1.17 \[ \frac{128}{3} e^4 x^9 \left (64 a^2 e^6-32 a d^4 e^3+d^8\right )-4 d e^3 x^8 \left (-1536 a^2 e^6+192 a d^4 e^3+d^8\right )-\frac{32}{7} d^2 e^2 x^7 \left (-768 a^2 e^6-24 a d^4 e^3+d^8\right )+\frac{1}{5} x^5 \left (-6144 a^2 d^4 e^6+16384 a^3 e^9+d^{12}\right )+8 a d e^2 x^4 \left (512 a^2 e^6-d^8\right )+128 a^2 d^6 e^4 x^3-1024 a^3 d^3 e^6 x^2+4096 a^4 e^8 x+\frac{2048}{13} e^8 x^{13} \left (8 a e^3-d^4\right )-512 d e^7 x^{12} \left (d^4-8 a e^3\right )+\frac{128}{11} d^2 e^6 x^{11} \left (384 a e^3-13 d^4\right )+\frac{128}{5} d^3 e^5 x^{10} \left (40 a e^3+3 d^4\right )-128 a d^3 e^4 x^6 \left (8 a e^3-d^4\right )+\frac{8192}{5} d^2 e^{10} x^{15}+1024 d^3 e^9 x^{14}+1024 d e^{11} x^{16}+\frac{4096 e^{12} x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 - 61
44*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 + 40*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

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Maple [A]  time = 0.003, size = 500, normalized size = 1.7 \begin{align*}{\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{ \left ( 16384\,a{e}^{5}-2048\,{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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+128/13*(128*a*e^5-16*d^4*e^2)*e^6*x^
13+1/12*(16384*a*e^10*d+256*(128*a*e^5-16*d^4*e^2)*d*e^5-2048*d^5*e^7)*x^12+1/11*(384*d^6*e^6+32768*a*e^9*d^2+
128*(128*a*e^5-16*d^4*e^2)*d^2*e^4)*x^11+1/10*(14336*a*e^8*d^3+256*d^7*e^5-32*(128*a*e^5-16*d^4*e^2)*d^3*e^3)*
x^10+1/9*(8192*a^2*e^10-8192*a*e^7*d^4+128*d^8*e^4+(128*a*e^5-16*d^4*e^2)^2)*x^9+1/8*(16384*a^2*e^9*d-2048*a*e
^6*d^5-32*d^9*e^3+256*a*e^4*d*(128*a*e^5-16*d^4*e^2))*x^8+1/7*(24576*a^2*e^8*d^2+512*a*e^5*d^6+2*d^6*(128*a*e^
5-16*d^4*e^2))*x^7+1/6*(-2048*a^2*e^7*d^3-32*a*e^2*d^3*(128*a*e^5-16*d^4*e^2)+256*d^7*a*e^4)*x^6+1/5*(128*a^2*
e^4*(128*a*e^5-16*d^4*e^2)-4096*a^2*e^6*d^4+d^12)*x^5+1/4*(16384*a^3*d*e^8-32*a*d^9*e^2)*x^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

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Maxima [A]  time = 1.02578, size = 517, normalized size = 1.75 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 838
6560*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

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Fricas [A]  time = 1.13587, size = 884, normalized size = 3. \begin{align*} \frac{4096}{17} x^{17} e^{12} + 1024 x^{16} e^{11} d + \frac{8192}{5} x^{15} e^{10} d^{2} + 1024 x^{14} e^{9} d^{3} - \frac{2048}{13} x^{13} e^{8} d^{4} + \frac{16384}{13} x^{13} e^{11} a - 512 x^{12} e^{7} d^{5} + 4096 x^{12} e^{10} d a - \frac{1664}{11} x^{11} e^{6} d^{6} + \frac{49152}{11} x^{11} e^{9} d^{2} a + \frac{384}{5} x^{10} e^{5} d^{7} + 1024 x^{10} e^{8} d^{3} a + \frac{128}{3} x^{9} e^{4} d^{8} - \frac{4096}{3} x^{9} e^{7} d^{4} a + \frac{8192}{3} x^{9} e^{10} a^{2} - 4 x^{8} e^{3} d^{9} - 768 x^{8} e^{6} d^{5} a + 6144 x^{8} e^{9} d a^{2} - \frac{32}{7} x^{7} e^{2} d^{10} + \frac{768}{7} x^{7} e^{5} d^{6} a + \frac{24576}{7} x^{7} e^{8} d^{2} a^{2} + 128 x^{6} e^{4} d^{7} a - 1024 x^{6} e^{7} d^{3} a^{2} + \frac{1}{5} x^{5} d^{12} - \frac{6144}{5} x^{5} e^{6} d^{4} a^{2} + \frac{16384}{5} x^{5} e^{9} a^{3} - 8 x^{4} e^{2} d^{9} a + 4096 x^{4} e^{8} d a^{3} + 128 x^{3} e^{4} d^{6} a^{2} - 1024 x^{2} e^{6} d^{3} a^{3} + 4096 x e^{8} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.124367, size = 366, normalized size = 1.24 \begin{align*} 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} \left (\frac{16384 a e^{11}}{13} - \frac{2048 d^{4} e^{8}}{13}\right ) + x^{12} \left (4096 a d e^{10} - 512 d^{5} e^{7}\right ) + x^{11} \left (\frac{49152 a d^{2} e^{9}}{11} - \frac{1664 d^{6} e^{6}}{11}\right ) + x^{10} \left (1024 a d^{3} e^{8} + \frac{384 d^{7} e^{5}}{5}\right ) + x^{9} \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} \left (6144 a^{2} d e^{9} - 768 a d^{5} e^{6} - 4 d^{9} e^{3}\right ) + x^{7} \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} \left (\frac{16384 a^{3} e^{9}}{5} - \frac{6144 a^{2} d^{4} e^{6}}{5} + \frac{d^{12}}{5}\right ) + x^{4} \left (4096 a^{3} d e^{8} - 8 a d^{9} e^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 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*(40
96*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 + 3
84*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 - 7
68*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*(-1
024*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*(409
6*a**3*d*e**8 - 8*a*d**9*e**2)

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Giac [A]  time = 1.11671, size = 436, normalized size = 1.48 \begin{align*} \frac{4096}{17} \, x^{17} e^{12} + 1024 \, d x^{16} e^{11} + \frac{8192}{5} \, d^{2} x^{15} e^{10} + 1024 \, d^{3} x^{14} e^{9} - \frac{2048}{13} \, d^{4} x^{13} e^{8} - 512 \, d^{5} x^{12} e^{7} - \frac{1664}{11} \, d^{6} x^{11} e^{6} + \frac{384}{5} \, d^{7} x^{10} e^{5} + \frac{128}{3} \, d^{8} x^{9} e^{4} - 4 \, d^{9} x^{8} e^{3} - \frac{32}{7} \, d^{10} x^{7} e^{2} + \frac{1}{5} \, d^{12} x^{5} + \frac{16384}{13} \, a x^{13} e^{11} + 4096 \, a d x^{12} e^{10} + \frac{49152}{11} \, a d^{2} x^{11} e^{9} + 1024 \, a d^{3} x^{10} e^{8} - \frac{4096}{3} \, a d^{4} x^{9} e^{7} - 768 \, a d^{5} x^{8} e^{6} + \frac{768}{7} \, a d^{6} x^{7} e^{5} + 128 \, a d^{7} x^{6} e^{4} - 8 \, a d^{9} x^{4} e^{2} + \frac{8192}{3} \, a^{2} x^{9} e^{10} + 6144 \, a^{2} d x^{8} e^{9} + \frac{24576}{7} \, a^{2} d^{2} x^{7} e^{8} - 1024 \, a^{2} d^{3} x^{6} e^{7} - \frac{6144}{5} \, a^{2} d^{4} x^{5} e^{6} + 128 \, a^{2} d^{6} x^{3} e^{4} + \frac{16384}{5} \, a^{3} x^{5} e^{9} + 4096 \, a^{3} d x^{4} e^{8} - 1024 \, a^{3} d^{3} x^{2} e^{6} + 4096 \, a^{4} x e^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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