\(\int (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4)^3 \, dx\) [40]

Optimal result

Integrand size = 32, antiderivative size = 203 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4-\frac {384}{5} a e^4 \left (d^4-4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac {24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8-\frac {128}{3} e^5 \left (d^4-4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]

[Out]

512*a^3*e^6*x-96*a^2*d^3*e^4*x^2+8*a*d^6*e^2*x^3-1/4*d*(-1536*a^2*e^6+d^8)*x^4-384/5*a*e^4*(-4*a*e^3+d^4)*x^5+
4*d^3*e^2*(-16*a*e^3+d^4)*x^6+24/7*d^2*e^3*(64*a*e^3+d^4)*x^7-24*d*e^4*(-16*a*e^3+d^4)*x^8-128/3*e^5*(-4*a*e^3
+d^4)*x^9+32*d^3*e^6*x^10+1536/11*d^2*e^7*x^11+128*d*e^8*x^12+512/13*e^9*x^13

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2086} \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^3 e^6 x-\frac {1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {128}{3} e^5 x^9 \left (d^4-4 a e^3\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )-\frac {384}{5} a e^4 x^5 \left (d^4-4 a e^3\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )+\frac {24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]

[In]

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

[Out]

512*a^3*e^6*x - 96*a^2*d^3*e^4*x^2 + 8*a*d^6*e^2*x^3 - (d*(d^8 - 1536*a^2*e^6)*x^4)/4 - (384*a*e^4*(d^4 - 4*a*
e^3)*x^5)/5 + 4*d^3*e^2*(d^4 - 16*a*e^3)*x^6 + (24*d^2*e^3*(d^4 + 64*a*e^3)*x^7)/7 - 24*d*e^4*(d^4 - 16*a*e^3)
*x^8 - (128*e^5*(d^4 - 4*a*e^3)*x^9)/3 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 128*d*e^8*x^12 + (512*e^9*
x^13)/13

Rule 2086

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (512 a^3 e^6-192 a^2 d^3 e^4 x+24 a d^6 e^2 x^2-d \left (d^8-1536 a^2 e^6\right ) x^3-384 a e^4 \left (d^4-4 a e^3\right ) x^4+24 d^3 e^2 \left (d^4-16 a e^3\right ) x^5+24 d^2 e^3 \left (d^4+64 a e^3\right ) x^6-192 d e^4 \left (d^4-16 a e^3\right ) x^7-384 e^5 \left (d^4-4 a e^3\right ) x^8+320 d^3 e^6 x^9+1536 d^2 e^7 x^{10}+1536 d e^8 x^{11}+512 e^9 x^{12}\right ) \, dx \\ & = 512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4-\frac {384}{5} a e^4 \left (d^4-4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac {24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8-\frac {128}{3} e^5 \left (d^4-4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.02 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4+\frac {384}{5} a e^4 \left (-d^4+4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac {24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8+\frac {128}{3} e^5 \left (-d^4+4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]

[In]

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

[Out]

512*a^3*e^6*x - 96*a^2*d^3*e^4*x^2 + 8*a*d^6*e^2*x^3 - (d*(d^8 - 1536*a^2*e^6)*x^4)/4 + (384*a*e^4*(-d^4 + 4*a
*e^3)*x^5)/5 + 4*d^3*e^2*(d^4 - 16*a*e^3)*x^6 + (24*d^2*e^3*(d^4 + 64*a*e^3)*x^7)/7 - 24*d*e^4*(d^4 - 16*a*e^3
)*x^8 + (128*e^5*(-d^4 + 4*a*e^3)*x^9)/3 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 128*d*e^8*x^12 + (512*e^
9*x^13)/13

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {512 e^{9} x^{13}}{13}+128 d \,e^{8} x^{12}+\frac {1536 d^{2} e^{7} x^{11}}{11}+32 d^{3} e^{6} x^{10}+\left (\frac {512}{3} a \,e^{8}-\frac {128}{3} d^{4} e^{5}\right ) x^{9}+\left (384 a \,e^{7} d -24 d^{5} e^{4}\right ) x^{8}+\left (\frac {1536}{7} a \,e^{6} d^{2}+\frac {24}{7} d^{6} e^{3}\right ) x^{7}+\left (-64 a \,e^{5} d^{3}+4 d^{7} e^{2}\right ) x^{6}+\left (\frac {1536}{5} a^{2} e^{7}-\frac {384}{5} a \,d^{4} e^{4}\right ) x^{5}+\left (384 a^{2} e^{6} d -\frac {1}{4} d^{9}\right ) x^{4}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x\)	\(200\)
gosper	\(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} a^{2} e^{7}-\frac {384}{5} x^{5} a \,d^{4} e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x\)	\(206\)
risch	\(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} a^{2} e^{7}-\frac {384}{5} x^{5} a \,d^{4} e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x\)	\(206\)
parallelrisch	\(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} a^{2} e^{7}-\frac {384}{5} x^{5} a \,d^{4} e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x\)	\(206\)
default	\(\frac {512 e^{9} x^{13}}{13}+128 d \,e^{8} x^{12}+\frac {1536 d^{2} e^{7} x^{11}}{11}+32 d^{3} e^{6} x^{10}+\frac {\left (512 a \,e^{8}-256 d^{4} e^{5}+8 e^{3} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{9}}{9}+\frac {\left (2048 a \,e^{7} d -64 d^{5} e^{4}+8 d \,e^{2} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{8}}{8}+\frac {\left (1536 a \,e^{6} d^{2}+24 d^{6} e^{3}\right ) x^{7}}{7}+\frac {\left (-256 a \,e^{5} d^{3}-d^{3} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )+8 d^{7} e^{2}\right ) x^{6}}{6}+\frac {\left (8 a \,e^{2} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )-256 a \,d^{4} e^{4}+512 a^{2} e^{7}\right ) x^{5}}{5}+\frac {\left (1536 a^{2} e^{6} d -d^{9}\right ) x^{4}}{4}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x\)	\(288\)

[In]

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

[Out]

512/13*e^9*x^13+128*d*e^8*x^12+1536/11*d^2*e^7*x^11+32*d^3*e^6*x^10+(512/3*a*e^8-128/3*d^4*e^5)*x^9+(384*a*d*e
^7-24*d^5*e^4)*x^8+(1536/7*a*e^6*d^2+24/7*d^6*e^3)*x^7+(-64*a*d^3*e^5+4*d^7*e^2)*x^6+(1536/5*a^2*e^7-384/5*a*d
^4*e^4)*x^5+(384*a^2*e^6*d-1/4*d^9)*x^4+8*a*d^6*e^2*x^3-96*a^2*d^3*e^4*x^2+512*a^3*e^6*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.98 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + 32 \, d^{3} e^{6} x^{10} + 8 \, a d^{6} e^{2} x^{3} - 96 \, a^{2} d^{3} e^{4} x^{2} + 512 \, a^{3} e^{6} x - \frac {128}{3} \, {\left (d^{4} e^{5} - 4 \, a e^{8}\right )} x^{9} - 24 \, {\left (d^{5} e^{4} - 16 \, a d e^{7}\right )} x^{8} + \frac {24}{7} \, {\left (d^{6} e^{3} + 64 \, a d^{2} e^{6}\right )} x^{7} + 4 \, {\left (d^{7} e^{2} - 16 \, a d^{3} e^{5}\right )} x^{6} - \frac {384}{5} \, {\left (a d^{4} e^{4} - 4 \, a^{2} e^{7}\right )} x^{5} - \frac {1}{4} \, {\left (d^{9} - 1536 \, a^{2} d e^{6}\right )} x^{4} \]

[In]

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

[Out]

512/13*e^9*x^13 + 128*d*e^8*x^12 + 1536/11*d^2*e^7*x^11 + 32*d^3*e^6*x^10 + 8*a*d^6*e^2*x^3 - 96*a^2*d^3*e^4*x
^2 + 512*a^3*e^6*x - 128/3*(d^4*e^5 - 4*a*e^8)*x^9 - 24*(d^5*e^4 - 16*a*d*e^7)*x^8 + 24/7*(d^6*e^3 + 64*a*d^2*
e^6)*x^7 + 4*(d^7*e^2 - 16*a*d^3*e^5)*x^6 - 384/5*(a*d^4*e^4 - 4*a^2*e^7)*x^5 - 1/4*(d^9 - 1536*a^2*d*e^6)*x^4

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.07 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^{3} e^{6} x - 96 a^{2} d^{3} e^{4} x^{2} + 8 a d^{6} e^{2} x^{3} + 32 d^{3} e^{6} x^{10} + \frac {1536 d^{2} e^{7} x^{11}}{11} + 128 d e^{8} x^{12} + \frac {512 e^{9} x^{13}}{13} + x^{9} \cdot \left (\frac {512 a e^{8}}{3} - \frac {128 d^{4} e^{5}}{3}\right ) + x^{8} \cdot \left (384 a d e^{7} - 24 d^{5} e^{4}\right ) + x^{7} \cdot \left (\frac {1536 a d^{2} e^{6}}{7} + \frac {24 d^{6} e^{3}}{7}\right ) + x^{6} \left (- 64 a d^{3} e^{5} + 4 d^{7} e^{2}\right ) + x^{5} \cdot \left (\frac {1536 a^{2} e^{7}}{5} - \frac {384 a d^{4} e^{4}}{5}\right ) + x^{4} \cdot \left (384 a^{2} d e^{6} - \frac {d^{9}}{4}\right ) \]

[In]

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

[Out]

512*a**3*e**6*x - 96*a**2*d**3*e**4*x**2 + 8*a*d**6*e**2*x**3 + 32*d**3*e**6*x**10 + 1536*d**2*e**7*x**11/11 +
128*d*e**8*x**12 + 512*e**9*x**13/13 + x**9*(512*a*e**8/3 - 128*d**4*e**5/3) + x**8*(384*a*d*e**7 - 24*d**5*e
**4) + x**7*(1536*a*d**2*e**6/7 + 24*d**6*e**3/7) + x**6*(-64*a*d**3*e**5 + 4*d**7*e**2) + x**5*(1536*a**2*e**
7/5 - 384*a*d**4*e**4/5) + x**4*(384*a**2*d*e**6 - d**9/4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.05 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + \frac {256}{5} \, d^{3} e^{6} x^{10} - \frac {1}{4} \, d^{9} x^{4} + 512 \, a^{3} e^{6} x + \frac {4}{7} \, {\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} + \frac {96}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{2} e^{4} - \frac {8}{15} \, {\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3} + \frac {8}{105} \, {\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\right )} a e^{2} \]

[In]

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

[Out]

512/13*e^9*x^13 + 128*d*e^8*x^12 + 1536/11*d^2*e^7*x^11 + 256/5*d^3*e^6*x^10 - 1/4*d^9*x^4 + 512*a^3*e^6*x + 4
/7*(6*e^3*x^7 + 7*d*e^2*x^6)*d^6 + 96/5*(16*e^3*x^5 + 20*d*e^2*x^4 - 5*d^3*x^2)*a^2*e^4 - 8/15*(36*e^6*x^10 +
80*d*e^5*x^9 + 45*d^2*e^4*x^8)*d^3 + 8/105*(2240*e^6*x^9 + 5040*d*e^5*x^8 + 2880*d^2*e^4*x^7 + 105*d^6*x^3 - 1
68*(5*e^3*x^6 + 6*d*e^2*x^5)*d^3)*a*e^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + 32 \, d^{3} e^{6} x^{10} - \frac {128}{3} \, d^{4} e^{5} x^{9} + \frac {512}{3} \, a e^{8} x^{9} - 24 \, d^{5} e^{4} x^{8} + 384 \, a d e^{7} x^{8} + \frac {24}{7} \, d^{6} e^{3} x^{7} + \frac {1536}{7} \, a d^{2} e^{6} x^{7} + 4 \, d^{7} e^{2} x^{6} - 64 \, a d^{3} e^{5} x^{6} - \frac {384}{5} \, a d^{4} e^{4} x^{5} + \frac {1536}{5} \, a^{2} e^{7} x^{5} - \frac {1}{4} \, d^{9} x^{4} + 384 \, a^{2} d e^{6} x^{4} + 8 \, a d^{6} e^{2} x^{3} - 96 \, a^{2} d^{3} e^{4} x^{2} + 512 \, a^{3} e^{6} x \]

[In]

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

[Out]

512/13*e^9*x^13 + 128*d*e^8*x^12 + 1536/11*d^2*e^7*x^11 + 32*d^3*e^6*x^10 - 128/3*d^4*e^5*x^9 + 512/3*a*e^8*x^
9 - 24*d^5*e^4*x^8 + 384*a*d*e^7*x^8 + 24/7*d^6*e^3*x^7 + 1536/7*a*d^2*e^6*x^7 + 4*d^7*e^2*x^6 - 64*a*d^3*e^5*
x^6 - 384/5*a*d^4*e^4*x^5 + 1536/5*a^2*e^7*x^5 - 1/4*d^9*x^4 + 384*a^2*d*e^6*x^4 + 8*a*d^6*e^2*x^3 - 96*a^2*d^
3*e^4*x^2 + 512*a^3*e^6*x

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.99 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512\,e^9\,x^{13}}{13}-x^4\,\left (\frac {d^9}{4}-384\,a^2\,d\,e^6\right )+\frac {128\,e^5\,x^9\,\left (4\,a\,e^3-d^4\right )}{3}+512\,a^3\,e^6\,x+128\,d\,e^8\,x^{12}+32\,d^3\,e^6\,x^{10}+\frac {1536\,d^2\,e^7\,x^{11}}{11}+8\,a\,d^6\,e^2\,x^3+\frac {384\,a\,e^4\,x^5\,\left (4\,a\,e^3-d^4\right )}{5}+24\,d\,e^4\,x^8\,\left (16\,a\,e^3-d^4\right )+\frac {24\,d^2\,e^3\,x^7\,\left (d^4+64\,a\,e^3\right )}{7}-96\,a^2\,d^3\,e^4\,x^2-4\,d^3\,e^2\,x^6\,\left (16\,a\,e^3-d^4\right ) \]

[In]

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

[Out]

(512*e^9*x^13)/13 - x^4*(d^9/4 - 384*a^2*d*e^6) + (128*e^5*x^9*(4*a*e^3 - d^4))/3 + 512*a^3*e^6*x + 128*d*e^8*
x^12 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 8*a*d^6*e^2*x^3 + (384*a*e^4*x^5*(4*a*e^3 - d^4))/5 + 24*d*e
^4*x^8*(16*a*e^3 - d^4) + (24*d^2*e^3*x^7*(64*a*e^3 + d^4))/7 - 96*a^2*d^3*e^4*x^2 - 4*d^3*e^2*x^6*(16*a*e^3 -
d^4)