Integrand size = 32, antiderivative size = 184 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {\left (5 d^4+256 a e^3\right )^3 x}{32768 e^3}-\frac {3 d^2 \left (5 d^4+256 a e^3\right )^2 (d+4 e x)^3}{65536 e^4}+\frac {3 \left (5 d^4+256 a e^3\right ) \left (41 d^4+256 a e^3\right ) (d+4 e x)^5}{655360 e^4}-\frac {9 d^2 \left (11 d^4+256 a e^3\right ) (d+4 e x)^7}{229376 e^4}+\frac {\left (41 d^4+256 a e^3\right ) (d+4 e x)^9}{393216 e^4}-\frac {9 d^2 (d+4 e x)^{11}}{720896 e^4}+\frac {(d+4 e x)^{13}}{1703936 e^4} \] Output:
1/32768*(256*a*e^3+5*d^4)^3*x/e^3-3/65536*d^2*(256*a*e^3+5*d^4)^2*(4*e*x+d )^3/e^4+3/655360*(256*a*e^3+5*d^4)*(256*a*e^3+41*d^4)*(4*e*x+d)^5/e^4-9/22 9376*d^2*(256*a*e^3+11*d^4)*(4*e*x+d)^7/e^4+1/393216*(256*a*e^3+41*d^4)*(4 *e*x+d)^9/e^4-9/720896*d^2*(4*e*x+d)^11/e^4+1/1703936*(4*e*x+d)^13/e^4
Time = 0.03 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.12 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4+\frac {384}{5} a e^4 \left (-d^4+4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac {24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8+\frac {128}{3} e^5 \left (-d^4+4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \] Input:
Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^3,x]
Output:
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
Time = 0.42 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.17, 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.
| \(\displaystyle \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4\right )^3d\left (\frac {d}{4 e}+x\right )\) |
| \(\Big \downarrow \) 1403 |
| \(\displaystyle \int \left (\frac {27}{32} d^4 e \left (256 a e^3+5 d^4\right ) \left (\frac {64 a e^3}{9 d^4}+\frac {41}{36}\right ) \left (\frac {d}{4 e}+x\right )^4+\frac {\left (256 a e^3+5 d^4\right )^3}{32768 e^3}+216 d^4 e^5 \left (\frac {64 a e^3}{9 d^4}+\frac {41}{36}\right ) \left (\frac {d}{4 e}+x\right )^8-27 d^6 e^3 \left (\frac {128 a e^3}{3 d^4}+\frac {11}{6}\right ) \left (\frac {d}{4 e}+x\right )^6-\frac {9 d^2 \left (256 a e^3+5 d^4\right )^2 \left (\frac {d}{4 e}+x\right )^2}{1024 e}-576 d^2 e^7 \left (\frac {d}{4 e}+x\right )^{10}+512 e^9 \left (\frac {d}{4 e}+x\right )^{12}\right )d\left (\frac {d}{4 e}+x\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{640} e \left (256 a e^3+5 d^4\right ) \left (256 a e^3+41 d^4\right ) \left (\frac {d}{4 e}+x\right )^5+\frac {\left (256 a e^3+5 d^4\right )^3 \left (\frac {d}{4 e}+x\right )}{32768 e^3}+\frac {2}{3} e^5 \left (256 a e^3+41 d^4\right ) \left (\frac {d}{4 e}+x\right )^9-\frac {9}{14} d^2 e^3 \left (256 a e^3+11 d^4\right ) \left (\frac {d}{4 e}+x\right )^7-\frac {3 d^2 \left (256 a e^3+5 d^4\right )^2 \left (\frac {d}{4 e}+x\right )^3}{1024 e}-\frac {576}{11} d^2 e^7 \left (\frac {d}{4 e}+x\right )^{11}+\frac {512}{13} e^9 \left (\frac {d}{4 e}+x\right )^{13}\) |
Input:
Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^3,x]
Output:
((5*d^4 + 256*a*e^3)^3*(d/(4*e) + x))/(32768*e^3) - (3*d^2*(5*d^4 + 256*a* e^3)^2*(d/(4*e) + x)^3)/(1024*e) + (3*e*(5*d^4 + 256*a*e^3)*(41*d^4 + 256* a*e^3)*(d/(4*e) + x)^5)/640 - (9*d^2*e^3*(11*d^4 + 256*a*e^3)*(d/(4*e) + x )^7)/14 + (2*e^5*(41*d^4 + 256*a*e^3)*(d/(4*e) + x)^9)/3 - (576*d^2*e^7*(d /(4*e) + x)^11)/11 + (512*e^9*(d/(4*e) + x)^13)/13
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]
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]
Time = 0.08 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09
| method | result | size |
| norman | \(\frac {512 e^{9} x^{13}}{13}+128 d \,e^{8} x^{12}+\frac {1536 d^{2} e^{7} x^{11}}{11}+32 d^{3} e^{6} x^{10}+\left (\frac {512}{3} a \,e^{8}-\frac {128}{3} d^{4} e^{5}\right ) x^{9}+\left (384 a \,e^{7} d -24 d^{5} e^{4}\right ) x^{8}+\left (\frac {1536}{7} a \,e^{6} d^{2}+\frac {24}{7} d^{6} e^{3}\right ) x^{7}+\left (-64 a \,e^{5} d^{3}+4 d^{7} e^{2}\right ) x^{6}+\left (\frac {1536}{5} e^{7} a^{2}-\frac {384}{5} d^{4} a \,e^{4}\right ) x^{5}+\left (384 a^{2} e^{6} d -\frac {1}{4} d^{9}\right ) x^{4}+8 a \,e^{2} d^{6} x^{3}-96 a^{2} e^{4} d^{3} x^{2}+512 a^{3} e^{6} x\) | \(200\) |
| gosper | \(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} e^{7} a^{2}-\frac {384}{5} x^{5} d^{4} a \,e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,e^{2} d^{6} x^{3}-96 a^{2} e^{4} d^{3} x^{2}+512 a^{3} e^{6} x\) | \(206\) |
| risch | \(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} e^{7} a^{2}-\frac {384}{5} x^{5} d^{4} a \,e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,e^{2} d^{6} x^{3}-96 a^{2} e^{4} d^{3} x^{2}+512 a^{3} e^{6} x\) | \(206\) |
| parallelrisch | \(\frac {512}{13} e^{9} x^{13}+128 d \,e^{8} x^{12}+\frac {1536}{11} d^{2} e^{7} x^{11}+32 d^{3} e^{6} x^{10}+\frac {512}{3} x^{9} a \,e^{8}-\frac {128}{3} x^{9} d^{4} e^{5}+384 a d \,e^{7} x^{8}-24 d^{5} e^{4} x^{8}+\frac {1536}{7} x^{7} a \,e^{6} d^{2}+\frac {24}{7} x^{7} d^{6} e^{3}-64 a \,d^{3} e^{5} x^{6}+4 d^{7} e^{2} x^{6}+\frac {1536}{5} x^{5} e^{7} a^{2}-\frac {384}{5} x^{5} d^{4} a \,e^{4}+384 x^{4} a^{2} e^{6} d -\frac {1}{4} x^{4} d^{9}+8 a \,e^{2} d^{6} x^{3}-96 a^{2} e^{4} d^{3} x^{2}+512 a^{3} e^{6} x\) | \(206\) |
| orering | \(\frac {x \left (2365440 e^{9} x^{12}+7687680 d \,e^{8} x^{11}+8386560 d^{2} e^{7} x^{10}+1921920 d^{3} e^{6} x^{9}+10250240 a \,e^{8} x^{8}-2562560 d^{4} e^{5} x^{8}+23063040 a d \,e^{7} x^{7}-1441440 d^{5} e^{4} x^{7}+13178880 a \,d^{2} e^{6} x^{6}+205920 d^{6} e^{3} x^{6}-3843840 a \,d^{3} e^{5} x^{5}+240240 d^{7} e^{2} x^{5}+18450432 a^{2} e^{7} x^{4}-4612608 a \,d^{4} e^{4} x^{4}+23063040 a^{2} d \,e^{6} x^{3}-15015 d^{9} x^{3}+480480 a \,e^{2} d^{6} x^{2}-5765760 a^{2} e^{4} d^{3} x +30750720 a^{3} e^{6}\right )}{60060}\) | \(206\) |
| default | \(\frac {512 e^{9} x^{13}}{13}+128 d \,e^{8} x^{12}+\frac {1536 d^{2} e^{7} x^{11}}{11}+32 d^{3} e^{6} x^{10}+\frac {\left (512 a \,e^{8}-256 d^{4} e^{5}+8 e^{3} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{9}}{9}+\frac {\left (2048 a \,e^{7} d -64 d^{5} e^{4}+8 d \,e^{2} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )\right ) x^{8}}{8}+\frac {\left (1536 a \,e^{6} d^{2}+24 d^{6} e^{3}\right ) x^{7}}{7}+\frac {\left (-256 a \,e^{5} d^{3}-d^{3} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )+8 d^{7} e^{2}\right ) x^{6}}{6}+\frac {\left (8 a \,e^{2} \left (128 a \,e^{5}-16 d^{4} e^{2}\right )-256 d^{4} a \,e^{4}+512 e^{7} a^{2}\right ) x^{5}}{5}+\frac {\left (1536 a^{2} e^{6} d -d^{9}\right ) x^{4}}{4}+8 a \,e^{2} d^{6} x^{3}-96 a^{2} e^{4} d^{3} x^{2}+512 a^{3} e^{6} x\) | \(288\) |
Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^3,x,method=_RETURNVERBOSE)
Output:
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+2 4/7*d^6*e^3)*x^7+(-64*a*d^3*e^5+4*d^7*e^2)*x^6+(1536/5*e^7*a^2-384/5*d^4*a *e^4)*x^5+(384*a^2*e^6*d-1/4*d^9)*x^4+8*a*e^2*d^6*x^3-96*a^2*e^4*d^3*x^2+5 12*a^3*e^6*x
Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + 32 \, d^{3} e^{6} x^{10} + 8 \, a d^{6} e^{2} x^{3} - 96 \, a^{2} d^{3} e^{4} x^{2} + 512 \, a^{3} e^{6} x - \frac {128}{3} \, {\left (d^{4} e^{5} - 4 \, a e^{8}\right )} x^{9} - 24 \, {\left (d^{5} e^{4} - 16 \, a d e^{7}\right )} x^{8} + \frac {24}{7} \, {\left (d^{6} e^{3} + 64 \, a d^{2} e^{6}\right )} x^{7} + 4 \, {\left (d^{7} e^{2} - 16 \, a d^{3} e^{5}\right )} x^{6} - \frac {384}{5} \, {\left (a d^{4} e^{4} - 4 \, a^{2} e^{7}\right )} x^{5} - \frac {1}{4} \, {\left (d^{9} - 1536 \, a^{2} d e^{6}\right )} x^{4} \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^3,x, algorithm="fricas")
Output:
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
Time = 0.05 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.18 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=512 a^{3} e^{6} x - 96 a^{2} d^{3} e^{4} x^{2} + 8 a d^{6} e^{2} x^{3} + 32 d^{3} e^{6} x^{10} + \frac {1536 d^{2} e^{7} x^{11}}{11} + 128 d e^{8} x^{12} + \frac {512 e^{9} x^{13}}{13} + x^{9} \cdot \left (\frac {512 a e^{8}}{3} - \frac {128 d^{4} e^{5}}{3}\right ) + x^{8} \cdot \left (384 a d e^{7} - 24 d^{5} e^{4}\right ) + x^{7} \cdot \left (\frac {1536 a d^{2} e^{6}}{7} + \frac {24 d^{6} e^{3}}{7}\right ) + x^{6} \left (- 64 a d^{3} e^{5} + 4 d^{7} e^{2}\right ) + x^{5} \cdot \left (\frac {1536 a^{2} e^{7}}{5} - \frac {384 a d^{4} e^{4}}{5}\right ) + x^{4} \cdot \left (384 a^{2} d e^{6} - \frac {d^{9}}{4}\right ) \] Input:
integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**3,x)
Output:
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)
Time = 0.04 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.16 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + \frac {256}{5} \, d^{3} e^{6} x^{10} - \frac {1}{4} \, d^{9} x^{4} + 512 \, a^{3} e^{6} x + \frac {4}{7} \, {\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} + \frac {96}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{2} e^{4} - \frac {8}{15} \, {\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3} + \frac {8}{105} \, {\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\right )} a e^{2} \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^3,x, algorithm="maxima")
Output:
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 - 168*(5*e^3*x^6 + 6*d*e^2*x^5)*d^3)*a*e^2
Time = 0.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + 32 \, d^{3} e^{6} x^{10} - \frac {128}{3} \, d^{4} e^{5} x^{9} + \frac {512}{3} \, a e^{8} x^{9} - 24 \, d^{5} e^{4} x^{8} + 384 \, a d e^{7} x^{8} + \frac {24}{7} \, d^{6} e^{3} x^{7} + \frac {1536}{7} \, a d^{2} e^{6} x^{7} + 4 \, d^{7} e^{2} x^{6} - 64 \, a d^{3} e^{5} x^{6} - \frac {384}{5} \, a d^{4} e^{4} x^{5} + \frac {1536}{5} \, a^{2} e^{7} x^{5} - \frac {1}{4} \, d^{9} x^{4} + 384 \, a^{2} d e^{6} x^{4} + 8 \, a d^{6} e^{2} x^{3} - 96 \, a^{2} d^{3} e^{4} x^{2} + 512 \, a^{3} e^{6} x \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^3,x, algorithm="giac")
Output:
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
Time = 0.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {512\,e^9\,x^{13}}{13}-x^4\,\left (\frac {d^9}{4}-384\,a^2\,d\,e^6\right )+\frac {128\,e^5\,x^9\,\left (4\,a\,e^3-d^4\right )}{3}+512\,a^3\,e^6\,x+128\,d\,e^8\,x^{12}+32\,d^3\,e^6\,x^{10}+\frac {1536\,d^2\,e^7\,x^{11}}{11}+8\,a\,d^6\,e^2\,x^3+\frac {384\,a\,e^4\,x^5\,\left (4\,a\,e^3-d^4\right )}{5}+24\,d\,e^4\,x^8\,\left (16\,a\,e^3-d^4\right )+\frac {24\,d^2\,e^3\,x^7\,\left (d^4+64\,a\,e^3\right )}{7}-96\,a^2\,d^3\,e^4\,x^2-4\,d^3\,e^2\,x^6\,\left (16\,a\,e^3-d^4\right ) \] Input:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^3,x)
Output:
(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)
Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^3 \, dx=\frac {x \left (2365440 e^{9} x^{12}+7687680 d \,e^{8} x^{11}+8386560 d^{2} e^{7} x^{10}+1921920 d^{3} e^{6} x^{9}+10250240 a \,e^{8} x^{8}-2562560 d^{4} e^{5} x^{8}+23063040 a d \,e^{7} x^{7}-1441440 d^{5} e^{4} x^{7}+13178880 a \,d^{2} e^{6} x^{6}+205920 d^{6} e^{3} x^{6}-3843840 a \,d^{3} e^{5} x^{5}+240240 d^{7} e^{2} x^{5}+18450432 a^{2} e^{7} x^{4}-4612608 a \,d^{4} e^{4} x^{4}+23063040 a^{2} d \,e^{6} x^{3}-15015 d^{9} x^{3}+480480 a \,d^{6} e^{2} x^{2}-5765760 a^{2} d^{3} e^{4} x +30750720 a^{3} e^{6}\right )}{60060} \] Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^3,x)
Output:
(x*(30750720*a**3*e**6 - 5765760*a**2*d**3*e**4*x + 23063040*a**2*d*e**6*x **3 + 18450432*a**2*e**7*x**4 + 480480*a*d**6*e**2*x**2 - 4612608*a*d**4*e **4*x**4 - 3843840*a*d**3*e**5*x**5 + 13178880*a*d**2*e**6*x**6 + 23063040 *a*d*e**7*x**7 + 10250240*a*e**8*x**8 - 15015*d**9*x**3 + 240240*d**7*e**2 *x**5 + 205920*d**6*e**3*x**6 - 1441440*d**5*e**4*x**7 - 2562560*d**4*e**5 *x**8 + 1921920*d**3*e**6*x**9 + 8386560*d**2*e**7*x**10 + 7687680*d*e**8* x**11 + 2365440*e**9*x**12))/60060