Integrand size = 97, antiderivative size = 26 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=e^{8 x} x \left (1-x+4 \left (16-e^4+x\right )^2\right )^2 \] Output:
(4*(x+16-exp(4))^2-x+1)^2*exp(4*x)^2*x
Time = 1.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=e^{8 x} x \left (1025+4 e^8+127 x+4 x^2-8 e^4 (16+x)\right )^2 \] Input:
Integrate[E^(8*x)*(1050625 + 8925700*x + 2155787*x^2 + 198696*x^3 + 8208*x ^4 + 128*x^5 + E^16*(16 + 128*x) + E^12*(-1024 - 8320*x - 512*x^2) + E^8*( 24584 + 202800*x + 24800*x^2 + 768*x^3) + E^4*(-262400 - 2197024*x - 40046 4*x^2 - 24704*x^3 - 512*x^4)),x]
Output:
E^(8*x)*x*(1025 + 4*E^8 + 127*x + 4*x^2 - 8*E^4*(16 + x))^2
Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(26)=52\).
Time = 0.83 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2626, 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.
| \(\displaystyle \int e^{8 x} \left (128 x^5+8208 x^4+198696 x^3+2155787 x^2+e^{12} \left (-512 x^2-8320 x-1024\right )+e^8 \left (768 x^3+24800 x^2+202800 x+24584\right )+e^4 \left (-512 x^4-24704 x^3-400464 x^2-2197024 x-262400\right )+8925700 x+e^{16} (128 x+16)+1050625\right ) \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (128 e^{8 x} x^5+8208 e^{8 x} x^4+198696 e^{8 x} x^3+2155787 e^{8 x} x^2-128 e^{8 x+12} \left (4 x^2+65 x+8\right )+8 e^{8 x+8} \left (96 x^3+3100 x^2+25350 x+3073\right )-16 e^{8 x+4} \left (32 x^4+1544 x^3+25029 x^2+137314 x+16400\right )+8925700 e^{8 x} x+1050625 e^{8 x}+16 e^{8 x+16} (8 x+1)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 e^{8 x} x^5+1016 e^{8 x} x^4-64 e^{8 x+4} x^4+24329 e^{8 x} x^3-3056 e^{8 x+4} x^3+96 e^{8 x+8} x^3+260350 e^{8 x} x^2-48912 e^{8 x+4} x^2+3064 e^{8 x+8} x^2-64 e^{8 x+12} x^2+1050625 e^{8 x} x-262400 e^{8 x+4} x+24584 e^{8 x+8} x-1024 e^{8 x+12} x-2 e^{8 x+16}+2 e^{8 x+16} (8 x+1)\) |
Input:
Int[E^(8*x)*(1050625 + 8925700*x + 2155787*x^2 + 198696*x^3 + 8208*x^4 + 1 28*x^5 + E^16*(16 + 128*x) + E^12*(-1024 - 8320*x - 512*x^2) + E^8*(24584 + 202800*x + 24800*x^2 + 768*x^3) + E^4*(-262400 - 2197024*x - 400464*x^2 - 24704*x^3 - 512*x^4)),x]
Output:
-2*E^(16 + 8*x) + 1050625*E^(8*x)*x - 262400*E^(4 + 8*x)*x + 24584*E^(8 + 8*x)*x - 1024*E^(12 + 8*x)*x + 260350*E^(8*x)*x^2 - 48912*E^(4 + 8*x)*x^2 + 3064*E^(8 + 8*x)*x^2 - 64*E^(12 + 8*x)*x^2 + 24329*E^(8*x)*x^3 - 3056*E^ (4 + 8*x)*x^3 + 96*E^(8 + 8*x)*x^3 + 1016*E^(8*x)*x^4 - 64*E^(4 + 8*x)*x^4 + 16*E^(8*x)*x^5 + 2*E^(16 + 8*x)*(1 + 8*x)
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38
| method | result | size |
| gosper | \(x \left (4 \,{\mathrm e}^{8}-8 x \,{\mathrm e}^{4}+4 x^{2}-128 \,{\mathrm e}^{4}+127 x +1025\right )^{2} {\mathrm e}^{8 x}\) | \(36\) |
| risch | \(\left (16 x \,{\mathrm e}^{16}-64 \,{\mathrm e}^{12} x^{2}+96 x^{3} {\mathrm e}^{8}-64 x^{4} {\mathrm e}^{4}+16 x^{5}-1024 x \,{\mathrm e}^{12}+3064 x^{2} {\mathrm e}^{8}-3056 x^{3} {\mathrm e}^{4}+1016 x^{4}+24584 x \,{\mathrm e}^{8}-48912 x^{2} {\mathrm e}^{4}+24329 x^{3}-262400 x \,{\mathrm e}^{4}+260350 x^{2}+1050625 x \right ) {\mathrm e}^{8 x}\) | \(92\) |
| norman | \(\left (-64 \,{\mathrm e}^{4}+1016\right ) x^{4} {\mathrm e}^{8 x}+\left (96 \,{\mathrm e}^{8}-3056 \,{\mathrm e}^{4}+24329\right ) x^{3} {\mathrm e}^{8 x}+\left (-64 \,{\mathrm e}^{12}+3064 \,{\mathrm e}^{8}-48912 \,{\mathrm e}^{4}+260350\right ) x^{2} {\mathrm e}^{8 x}+\left (-262400 \,{\mathrm e}^{4}+16 \,{\mathrm e}^{16}-1024 \,{\mathrm e}^{12}+24584 \,{\mathrm e}^{8}+1050625\right ) x \,{\mathrm e}^{8 x}+16 \,{\mathrm e}^{8 x} x^{5}\) | \(111\) |
| parallelrisch | \(16 \,{\mathrm e}^{8 x} {\mathrm e}^{16} x -64 \,{\mathrm e}^{8 x} {\mathrm e}^{12} x^{2}+96 \,{\mathrm e}^{8 x} {\mathrm e}^{8} x^{3}-64 \,{\mathrm e}^{8 x} {\mathrm e}^{4} x^{4}+16 \,{\mathrm e}^{8 x} x^{5}-1024 \,{\mathrm e}^{8 x} {\mathrm e}^{12} x +3064 \,{\mathrm e}^{8 x} {\mathrm e}^{8} x^{2}-3056 \,{\mathrm e}^{8 x} {\mathrm e}^{4} x^{3}+1016 \,{\mathrm e}^{8 x} x^{4}+24584 \,{\mathrm e}^{8 x} {\mathrm e}^{8} x -48912 \,{\mathrm e}^{8 x} {\mathrm e}^{4} x^{2}+24329 \,{\mathrm e}^{8 x} x^{3}-262400 \,{\mathrm e}^{8 x} {\mathrm e}^{4} x +260350 \,{\mathrm e}^{8 x} x^{2}+1050625 x \,{\mathrm e}^{8 x}\) | \(189\) |
| meijerg | \(-\frac {33619985}{256}-\frac {\left (-196608 x^{5}+122880 x^{4}-61440 x^{3}+23040 x^{2}-5760 x +720\right ) {\mathrm e}^{8 x}}{12288}-\frac {\left (-512 \,{\mathrm e}^{4}+8208\right ) \left (24-\frac {\left (20480 x^{4}-10240 x^{3}+3840 x^{2}-960 x +120\right ) {\mathrm e}^{8 x}}{5}\right )}{32768}+\frac {\left (768 \,{\mathrm e}^{8}-24704 \,{\mathrm e}^{4}+198696\right ) \left (6-\frac {\left (-2048 x^{3}+768 x^{2}-192 x +24\right ) {\mathrm e}^{8 x}}{4}\right )}{4096}-\frac {\left (-512 \,{\mathrm e}^{12}+24800 \,{\mathrm e}^{8}-400464 \,{\mathrm e}^{4}+2155787\right ) \left (2-\frac {\left (192 x^{2}-48 x +6\right ) {\mathrm e}^{8 x}}{3}\right )}{512}+\frac {\left (128 \,{\mathrm e}^{16}-8320 \,{\mathrm e}^{12}+202800 \,{\mathrm e}^{8}-2197024 \,{\mathrm e}^{4}+8925700\right ) \left (1-\frac {\left (2-16 x \right ) {\mathrm e}^{8 x}}{2}\right )}{64}-2 \,{\mathrm e}^{16} \left (1-{\mathrm e}^{8 x}\right )+128 \,{\mathrm e}^{12} \left (1-{\mathrm e}^{8 x}\right )-3073 \,{\mathrm e}^{8} \left (1-{\mathrm e}^{8 x}\right )+32800 \,{\mathrm e}^{4} \left (1-{\mathrm e}^{8 x}\right )+\frac {1050625 \,{\mathrm e}^{8 x}}{8}\) | \(226\) |
| orering | \(\frac {x \left (64 x^{3} {\mathrm e}^{4}-16 x^{4}+3056 x^{2} {\mathrm e}^{4}-96 x^{2} {\mathrm e}^{8}-1016 x^{3}+48912 x \,{\mathrm e}^{4}+64 x \,{\mathrm e}^{12}-3064 x \,{\mathrm e}^{8}-24329 x^{2}+262400 \,{\mathrm e}^{4}-16 \,{\mathrm e}^{16}+1024 \,{\mathrm e}^{12}-24584 \,{\mathrm e}^{8}-260350 x -1050625\right ) \left (\left (128 x +16\right ) {\mathrm e}^{16}+\left (-512 x^{2}-8320 x -1024\right ) {\mathrm e}^{12}+\left (768 x^{3}+24800 x^{2}+202800 x +24584\right ) {\mathrm e}^{8}+\left (-512 x^{4}-24704 x^{3}-400464 x^{2}-2197024 x -262400\right ) {\mathrm e}^{4}+128 x^{5}+8208 x^{4}+198696 x^{3}+2155787 x^{2}+8925700 x +1050625\right ) {\mathrm e}^{8 x}}{-128 x \,{\mathrm e}^{16}+512 \,{\mathrm e}^{12} x^{2}-768 x^{3} {\mathrm e}^{8}+512 x^{4} {\mathrm e}^{4}-128 x^{5}-16 \,{\mathrm e}^{16}+8320 x \,{\mathrm e}^{12}-24800 x^{2} {\mathrm e}^{8}+24704 x^{3} {\mathrm e}^{4}-8208 x^{4}+1024 \,{\mathrm e}^{12}-202800 x \,{\mathrm e}^{8}+400464 x^{2} {\mathrm e}^{4}-198696 x^{3}-24584 \,{\mathrm e}^{8}+2197024 x \,{\mathrm e}^{4}-2155787 x^{2}+262400 \,{\mathrm e}^{4}-8925700 x -1050625}\) | \(279\) |
| derivativedivides | \(1050625 x \,{\mathrm e}^{8 x}+260350 \,{\mathrm e}^{8 x} x^{2}+24329 \,{\mathrm e}^{8 x} x^{3}+1016 \,{\mathrm e}^{8 x} x^{4}+16 \,{\mathrm e}^{8 x} x^{5}-32800 \,{\mathrm e}^{8 x} {\mathrm e}^{4}+3073 \,{\mathrm e}^{8 x} {\mathrm e}^{8}-128 \,{\mathrm e}^{8 x} {\mathrm e}^{12}+2 \,{\mathrm e}^{8 x} {\mathrm e}^{16}-137314 \,{\mathrm e}^{4} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-\frac {25029 \,{\mathrm e}^{4} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{4}-\frac {193 \,{\mathrm e}^{4} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 \,{\mathrm e}^{8 x} x^{2}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )}{2}-\frac {{\mathrm e}^{4} \left (128 \,{\mathrm e}^{8 x} x^{4}-64 \,{\mathrm e}^{8 x} x^{3}+24 \,{\mathrm e}^{8 x} x^{2}-6 x \,{\mathrm e}^{8 x}+\frac {3 \,{\mathrm e}^{8 x}}{4}\right )}{2}+12675 \,{\mathrm e}^{8} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )+\frac {775 \,{\mathrm e}^{8} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{2}+3 \,{\mathrm e}^{8} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 \,{\mathrm e}^{8 x} x^{2}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )-520 \,{\mathrm e}^{12} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-8 \,{\mathrm e}^{12} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )+8 \,{\mathrm e}^{16} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )\) | \(443\) |
| default | \(1050625 x \,{\mathrm e}^{8 x}+260350 \,{\mathrm e}^{8 x} x^{2}+24329 \,{\mathrm e}^{8 x} x^{3}+1016 \,{\mathrm e}^{8 x} x^{4}+16 \,{\mathrm e}^{8 x} x^{5}-32800 \,{\mathrm e}^{8 x} {\mathrm e}^{4}+3073 \,{\mathrm e}^{8 x} {\mathrm e}^{8}-128 \,{\mathrm e}^{8 x} {\mathrm e}^{12}+2 \,{\mathrm e}^{8 x} {\mathrm e}^{16}-137314 \,{\mathrm e}^{4} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-\frac {25029 \,{\mathrm e}^{4} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{4}-\frac {193 \,{\mathrm e}^{4} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 \,{\mathrm e}^{8 x} x^{2}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )}{2}-\frac {{\mathrm e}^{4} \left (128 \,{\mathrm e}^{8 x} x^{4}-64 \,{\mathrm e}^{8 x} x^{3}+24 \,{\mathrm e}^{8 x} x^{2}-6 x \,{\mathrm e}^{8 x}+\frac {3 \,{\mathrm e}^{8 x}}{4}\right )}{2}+12675 \,{\mathrm e}^{8} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )+\frac {775 \,{\mathrm e}^{8} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )}{2}+3 \,{\mathrm e}^{8} \left (32 \,{\mathrm e}^{8 x} x^{3}-12 \,{\mathrm e}^{8 x} x^{2}+3 x \,{\mathrm e}^{8 x}-\frac {3 \,{\mathrm e}^{8 x}}{8}\right )-520 \,{\mathrm e}^{12} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )-8 \,{\mathrm e}^{12} \left (8 \,{\mathrm e}^{8 x} x^{2}-2 x \,{\mathrm e}^{8 x}+\frac {{\mathrm e}^{8 x}}{4}\right )+8 \,{\mathrm e}^{16} \left (2 x \,{\mathrm e}^{8 x}-\frac {{\mathrm e}^{8 x}}{4}\right )\) | \(443\) |
Input:
int(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^ 2+202800*x+24584)*exp(4)^2+(-512*x^4-24704*x^3-400464*x^2-2197024*x-262400 )*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)*exp(4* x)^2,x,method=_RETURNVERBOSE)
Output:
x*(4*exp(4)^2-8*x*exp(4)+4*x^2-128*exp(4)+127*x+1025)^2*exp(4*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.31 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx={\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 16 \, x e^{16} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{12} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{8} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{4} + 1050625 \, x\right )} e^{\left (8 \, x\right )} \] Input:
integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24 800*x^2+202800*x+24584)*exp(4)^2+(-512*x^4-24704*x^3-400464*x^2-2197024*x- 262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)* exp(4*x)^2,x, algorithm="fricas")
Output:
(16*x^5 + 1016*x^4 + 24329*x^3 + 260350*x^2 + 16*x*e^16 - 64*(x^2 + 16*x)* e^12 + 8*(12*x^3 + 383*x^2 + 3073*x)*e^8 - 16*(4*x^4 + 191*x^3 + 3057*x^2 + 16400*x)*e^4 + 1050625*x)*e^(8*x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.04 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=\left (16 x^{5} - 64 x^{4} e^{4} + 1016 x^{4} - 3056 x^{3} e^{4} + 24329 x^{3} + 96 x^{3} e^{8} - 64 x^{2} e^{12} - 48912 x^{2} e^{4} + 260350 x^{2} + 3064 x^{2} e^{8} - 1024 x e^{12} - 262400 x e^{4} + 1050625 x + 24584 x e^{8} + 16 x e^{16}\right ) e^{8 x} \] Input:
integrate(((128*x+16)*exp(4)**4+(-512*x**2-8320*x-1024)*exp(4)**3+(768*x** 3+24800*x**2+202800*x+24584)*exp(4)**2+(-512*x**4-24704*x**3-400464*x**2-2 197024*x-262400)*exp(4)+128*x**5+8208*x**4+198696*x**3+2155787*x**2+892570 0*x+1050625)*exp(4*x)**2,x)
Output:
(16*x**5 - 64*x**4*exp(4) + 1016*x**4 - 3056*x**3*exp(4) + 24329*x**3 + 96 *x**3*exp(8) - 64*x**2*exp(12) - 48912*x**2*exp(4) + 260350*x**2 + 3064*x* *2*exp(8) - 1024*x*exp(12) - 262400*x*exp(4) + 1050625*x + 24584*x*exp(8) + 16*x*exp(16))*exp(8*x)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (24) = 48\).
Time = 0.05 (sec) , antiderivative size = 368, normalized size of antiderivative = 14.15 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=\frac {1}{256} \, {\left (4096 \, x^{5} - 2560 \, x^{4} + 1280 \, x^{3} - 480 \, x^{2} + 120 \, x - 15\right )} e^{\left (8 \, x\right )} - \frac {1}{8} \, {\left (512 \, x^{4} e^{4} - 256 \, x^{3} e^{4} + 96 \, x^{2} e^{4} - 24 \, x e^{4} + 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {513}{256} \, {\left (512 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 24 \, x + 3\right )} e^{\left (8 \, x\right )} + \frac {3}{8} \, {\left (256 \, x^{3} e^{8} - 96 \, x^{2} e^{8} + 24 \, x e^{8} - 3 \, e^{8}\right )} e^{\left (8 \, x\right )} - \frac {193}{16} \, {\left (256 \, x^{3} e^{4} - 96 \, x^{2} e^{4} + 24 \, x e^{4} - 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {24837}{256} \, {\left (256 \, x^{3} - 96 \, x^{2} + 24 \, x - 3\right )} e^{\left (8 \, x\right )} - 2 \, {\left (32 \, x^{2} e^{12} - 8 \, x e^{12} + e^{12}\right )} e^{\left (8 \, x\right )} + \frac {775}{8} \, {\left (32 \, x^{2} e^{8} - 8 \, x e^{8} + e^{8}\right )} e^{\left (8 \, x\right )} - \frac {25029}{16} \, {\left (32 \, x^{2} e^{4} - 8 \, x e^{4} + e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2155787}{256} \, {\left (32 \, x^{2} - 8 \, x + 1\right )} e^{\left (8 \, x\right )} + 2 \, {\left (8 \, x e^{16} - e^{16}\right )} e^{\left (8 \, x\right )} - 130 \, {\left (8 \, x e^{12} - e^{12}\right )} e^{\left (8 \, x\right )} + \frac {12675}{4} \, {\left (8 \, x e^{8} - e^{8}\right )} e^{\left (8 \, x\right )} - \frac {68657}{2} \, {\left (8 \, x e^{4} - e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2231425}{16} \, {\left (8 \, x - 1\right )} e^{\left (8 \, x\right )} + \frac {1050625}{8} \, e^{\left (8 \, x\right )} + 2 \, e^{\left (8 \, x + 16\right )} - 128 \, e^{\left (8 \, x + 12\right )} + 3073 \, e^{\left (8 \, x + 8\right )} - 32800 \, e^{\left (8 \, x + 4\right )} \] Input:
integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24 800*x^2+202800*x+24584)*exp(4)^2+(-512*x^4-24704*x^3-400464*x^2-2197024*x- 262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)* exp(4*x)^2,x, algorithm="maxima")
Output:
1/256*(4096*x^5 - 2560*x^4 + 1280*x^3 - 480*x^2 + 120*x - 15)*e^(8*x) - 1/ 8*(512*x^4*e^4 - 256*x^3*e^4 + 96*x^2*e^4 - 24*x*e^4 + 3*e^4)*e^(8*x) + 51 3/256*(512*x^4 - 256*x^3 + 96*x^2 - 24*x + 3)*e^(8*x) + 3/8*(256*x^3*e^8 - 96*x^2*e^8 + 24*x*e^8 - 3*e^8)*e^(8*x) - 193/16*(256*x^3*e^4 - 96*x^2*e^4 + 24*x*e^4 - 3*e^4)*e^(8*x) + 24837/256*(256*x^3 - 96*x^2 + 24*x - 3)*e^( 8*x) - 2*(32*x^2*e^12 - 8*x*e^12 + e^12)*e^(8*x) + 775/8*(32*x^2*e^8 - 8*x *e^8 + e^8)*e^(8*x) - 25029/16*(32*x^2*e^4 - 8*x*e^4 + e^4)*e^(8*x) + 2155 787/256*(32*x^2 - 8*x + 1)*e^(8*x) + 2*(8*x*e^16 - e^16)*e^(8*x) - 130*(8* x*e^12 - e^12)*e^(8*x) + 12675/4*(8*x*e^8 - e^8)*e^(8*x) - 68657/2*(8*x*e^ 4 - e^4)*e^(8*x) + 2231425/16*(8*x - 1)*e^(8*x) + 1050625/8*e^(8*x) + 2*e^ (8*x + 16) - 128*e^(8*x + 12) + 3073*e^(8*x + 8) - 32800*e^(8*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.96 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx={\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 1050625 \, x\right )} e^{\left (8 \, x\right )} + 16 \, x e^{\left (8 \, x + 16\right )} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{\left (8 \, x + 12\right )} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{\left (8 \, x + 8\right )} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{\left (8 \, x + 4\right )} \] Input:
integrate(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24 800*x^2+202800*x+24584)*exp(4)^2+(-512*x^4-24704*x^3-400464*x^2-2197024*x- 262400)*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)* exp(4*x)^2,x, algorithm="giac")
Output:
(16*x^5 + 1016*x^4 + 24329*x^3 + 260350*x^2 + 1050625*x)*e^(8*x) + 16*x*e^ (8*x + 16) - 64*(x^2 + 16*x)*e^(8*x + 12) + 8*(12*x^3 + 383*x^2 + 3073*x)* e^(8*x + 8) - 16*(4*x^4 + 191*x^3 + 3057*x^2 + 16400*x)*e^(8*x + 4)
Time = 3.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=x\,{\mathrm {e}}^{8\,x}\,{\left (127\,x-128\,{\mathrm {e}}^4+4\,{\mathrm {e}}^8-8\,x\,{\mathrm {e}}^4+4\,x^2+1025\right )}^2 \] Input:
int(exp(8*x)*(8925700*x - exp(12)*(8320*x + 512*x^2 + 1024) + exp(8)*(2028 00*x + 24800*x^2 + 768*x^3 + 24584) - exp(4)*(2197024*x + 400464*x^2 + 247 04*x^3 + 512*x^4 + 262400) + 2155787*x^2 + 198696*x^3 + 8208*x^4 + 128*x^5 + exp(16)*(128*x + 16) + 1050625),x)
Output:
x*exp(8*x)*(127*x - 128*exp(4) + 4*exp(8) - 8*x*exp(4) + 4*x^2 + 1025)^2
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int e^{8 x} \left (1050625+8925700 x+2155787 x^2+198696 x^3+8208 x^4+128 x^5+e^{16} (16+128 x)+e^{12} \left (-1024-8320 x-512 x^2\right )+e^8 \left (24584+202800 x+24800 x^2+768 x^3\right )+e^4 \left (-262400-2197024 x-400464 x^2-24704 x^3-512 x^4\right )\right ) \, dx=e^{8 x} x \left (16 e^{16}-64 e^{12} x -1024 e^{12}+96 e^{8} x^{2}+3064 e^{8} x +24584 e^{8}-64 e^{4} x^{3}-3056 e^{4} x^{2}-48912 e^{4} x -262400 e^{4}+16 x^{4}+1016 x^{3}+24329 x^{2}+260350 x +1050625\right ) \] Input:
int(((128*x+16)*exp(4)^4+(-512*x^2-8320*x-1024)*exp(4)^3+(768*x^3+24800*x^ 2+202800*x+24584)*exp(4)^2+(-512*x^4-24704*x^3-400464*x^2-2197024*x-262400 )*exp(4)+128*x^5+8208*x^4+198696*x^3+2155787*x^2+8925700*x+1050625)*exp(4* x)^2,x)
Output:
e**(8*x)*x*(16*e**16 - 64*e**12*x - 1024*e**12 + 96*e**8*x**2 + 3064*e**8* x + 24584*e**8 - 64*e**4*x**3 - 3056*e**4*x**2 - 48912*e**4*x - 262400*e** 4 + 16*x**4 + 1016*x**3 + 24329*x**2 + 260350*x + 1050625)