\(\int (2 e^2 x+e^{1+3 x} (-64 x^3-48 x^4+96 x^5+48 x^6)+e^{8 x} (96 x^5+128 x^6)+e^{7 x} (384 x^5+448 x^6-512 x^7-448 x^8)+e^{1+2 x} (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8)+e^{6 x} (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10})+e^{5 x} (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12})+e^{4 x} (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e (-32 x^3-32 x^4))) \, dx\) [1782]

Optimal result

Integrand size = 273, antiderivative size = 32 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=x^2 \left (-e+4 e^{2 x} x^2 \left (1+e^x-x^2\right )^2\right )^2 \] Output:

(4*(-x^2+1+exp(x))^2*exp(x)^2*x^2-exp(1))^2*x^2
 

Mathematica [A] (verified)

Time = 10.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=\left (e x-4 e^{4 x} x^3+8 e^{3 x} x^3 \left (-1+x^2\right )-4 e^{2 x} x^3 \left (-1+x^2\right )^2\right )^2 \] Input:

Integrate[2*E^2*x + E^(1 + 3*x)*(-64*x^3 - 48*x^4 + 96*x^5 + 48*x^6) + E^( 
8*x)*(96*x^5 + 128*x^6) + E^(7*x)*(384*x^5 + 448*x^6 - 512*x^7 - 448*x^8) 
+ E^(1 + 2*x)*(-32*x^3 - 16*x^4 + 96*x^5 + 32*x^6 - 64*x^7 - 16*x^8) + E^( 
6*x)*(576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x^9 + 576*x^10) + E^(5 
*x)*(384*x^5 + 320*x^6 - 1536*x^7 - 960*x^8 + 1920*x^9 + 960*x^10 - 768*x^ 
11 - 320*x^12) + E^(4*x)*(96*x^5 + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^9 + 
384*x^10 - 768*x^11 - 256*x^12 + 224*x^13 + 64*x^14 + E*(-32*x^3 - 32*x^4) 
),x]
 

Output:

(E*x - 4*E^(4*x)*x^3 + 8*E^(3*x)*x^3*(-1 + x^2) - 4*E^(2*x)*x^3*(-1 + x^2) 
^2)^2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(230\) vs. \(2(32)=64\).

Time = 4.03 (sec) , antiderivative size = 230, normalized size of antiderivative = 7.19, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {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.

\(\displaystyle \int \left (e^{8 x} \left (128 x^6+96 x^5\right )+e^{7 x} \left (-448 x^8-512 x^7+448 x^6+384 x^5\right )+e^{3 x+1} \left (48 x^6+96 x^5-48 x^4-64 x^3\right )+e^{6 x} \left (576 x^{10}+960 x^9-1152 x^8-1536 x^7+576 x^6+576 x^5\right )+e^{2 x+1} \left (-16 x^8-64 x^7+32 x^6+96 x^5-16 x^4-32 x^3\right )+e^{5 x} \left (-320 x^{12}-768 x^{11}+960 x^{10}+1920 x^9-960 x^8-1536 x^7+320 x^6+384 x^5\right )+2 e^2 x+e^{4 x} \left (64 x^{14}+224 x^{13}-256 x^{12}-768 x^{11}+384 x^{10}+960 x^9-256 x^8-512 x^7+64 x^6+96 x^5+e \left (-32 x^4-32 x^3\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 16 e^{4 x} x^{14}-64 e^{4 x} x^{12}-64 e^{5 x} x^{12}+96 e^{4 x} x^{10}+192 e^{5 x} x^{10}+96 e^{6 x} x^{10}-64 e^{4 x} x^8-192 e^{5 x} x^8-192 e^{6 x} x^8-64 e^{7 x} x^8-8 e^{2 x+1} x^8+16 e^{4 x} x^6+64 e^{5 x} x^6+96 e^{6 x} x^6+64 e^{7 x} x^6+16 e^{8 x} x^6+16 e^{2 x+1} x^6+16 e^{3 x+1} x^6-8 e^{2 x+1} x^4-16 e^{3 x+1} x^4-8 e^{4 x+1} x^4+e^2 x^2\)

Input:

Int[2*E^2*x + E^(1 + 3*x)*(-64*x^3 - 48*x^4 + 96*x^5 + 48*x^6) + E^(8*x)*( 
96*x^5 + 128*x^6) + E^(7*x)*(384*x^5 + 448*x^6 - 512*x^7 - 448*x^8) + E^(1 
 + 2*x)*(-32*x^3 - 16*x^4 + 96*x^5 + 32*x^6 - 64*x^7 - 16*x^8) + E^(6*x)*( 
576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x^9 + 576*x^10) + E^(5*x)*(3 
84*x^5 + 320*x^6 - 1536*x^7 - 960*x^8 + 1920*x^9 + 960*x^10 - 768*x^11 - 3 
20*x^12) + E^(4*x)*(96*x^5 + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^9 + 384*x^ 
10 - 768*x^11 - 256*x^12 + 224*x^13 + 64*x^14 + E*(-32*x^3 - 32*x^4)),x]
 

Output:

E^2*x^2 - 8*E^(1 + 2*x)*x^4 - 16*E^(1 + 3*x)*x^4 - 8*E^(1 + 4*x)*x^4 + 16* 
E^(4*x)*x^6 + 64*E^(5*x)*x^6 + 96*E^(6*x)*x^6 + 64*E^(7*x)*x^6 + 16*E^(8*x 
)*x^6 + 16*E^(1 + 2*x)*x^6 + 16*E^(1 + 3*x)*x^6 - 64*E^(4*x)*x^8 - 192*E^( 
5*x)*x^8 - 192*E^(6*x)*x^8 - 64*E^(7*x)*x^8 - 8*E^(1 + 2*x)*x^8 + 96*E^(4* 
x)*x^10 + 192*E^(5*x)*x^10 + 96*E^(6*x)*x^10 - 64*E^(4*x)*x^12 - 64*E^(5*x 
)*x^12 + 16*E^(4*x)*x^14
 

Defintions of rubi rules used

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(31)=62\).

Time = 9.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.97

method	result	size
risch	\(16 \,{\mathrm e}^{8 x} x^{6}+\left (-64 x^{8}+64 x^{6}\right ) {\mathrm e}^{7 x}+\left (96 x^{10}-192 x^{8}+96 x^{6}\right ) {\mathrm e}^{6 x}+\left (-64 x^{12}+192 x^{10}-192 x^{8}+64 x^{6}\right ) {\mathrm e}^{5 x}+\left (16 x^{14}-64 x^{12}+96 x^{10}-64 x^{8}+16 x^{6}-8 x^{4} {\mathrm e}\right ) {\mathrm e}^{4 x}+\left (16 x^{6}-16 x^{4}\right ) {\mathrm e}^{1+3 x}+\left (-8 x^{8}+16 x^{6}-8 x^{4}\right ) {\mathrm e}^{1+2 x}+x^{2} {\mathrm e}^{2}\)	\(159\)
parallelrisch	\(16 \,{\mathrm e}^{4 x} x^{14}-64 \,{\mathrm e}^{5 x} x^{12}+96 \,{\mathrm e}^{6 x} x^{10}-64 \,{\mathrm e}^{4 x} x^{12}-64 \,{\mathrm e}^{7 x} x^{8}+192 \,{\mathrm e}^{5 x} x^{10}+16 \,{\mathrm e}^{8 x} x^{6}-192 \,{\mathrm e}^{6 x} x^{8}+96 \,{\mathrm e}^{4 x} x^{10}+64 \,{\mathrm e}^{7 x} x^{6}-192 \,{\mathrm e}^{5 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{6}-64 \,{\mathrm e}^{4 x} x^{8}+64 x^{6} {\mathrm e}^{5 x}-8 \,{\mathrm e}^{2 x} {\mathrm e} x^{8}+16 \,{\mathrm e}^{4 x} x^{6}+16 \,{\mathrm e}^{3 x} {\mathrm e} x^{6}-8 \,{\mathrm e}^{4 x} {\mathrm e} x^{4}+16 \,{\mathrm e}^{2 x} {\mathrm e} x^{6}-16 \,{\mathrm e}^{3 x} {\mathrm e} x^{4}-8 \,{\mathrm e}^{2 x} {\mathrm e} x^{4}+x^{2} {\mathrm e}^{2}\)	\(211\)
default	\(-64 \,{\mathrm e}^{4 x} x^{12}+96 \,{\mathrm e}^{4 x} x^{10}-64 \,{\mathrm e}^{4 x} x^{8}+16 \,{\mathrm e}^{4 x} x^{6}+16 \,{\mathrm e}^{4 x} x^{14}-32 \,{\mathrm e} \left (\frac {x^{3} {\mathrm e}^{4 x}}{4}-\frac {3 x^{2} {\mathrm e}^{4 x}}{16}+\frac {3 x \,{\mathrm e}^{4 x}}{32}-\frac {3 \,{\mathrm e}^{4 x}}{128}\right )-32 \,{\mathrm e} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}-\frac {x^{3} {\mathrm e}^{4 x}}{4}+\frac {3 x^{2} {\mathrm e}^{4 x}}{16}-\frac {3 x \,{\mathrm e}^{4 x}}{32}+\frac {3 \,{\mathrm e}^{4 x}}{128}\right )+16 \,{\mathrm e}^{8 x} x^{6}+64 \,{\mathrm e}^{7 x} x^{6}-64 \,{\mathrm e}^{7 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{6}-192 \,{\mathrm e}^{6 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{10}-64 \,{\mathrm e}^{5 x} x^{12}+192 \,{\mathrm e}^{5 x} x^{10}-192 \,{\mathrm e}^{5 x} x^{8}+64 x^{6} {\mathrm e}^{5 x}+16 \,{\mathrm e} \left (-x^{4} {\mathrm e}^{3 x}+x^{6} {\mathrm e}^{3 x}\right )+16 \,{\mathrm e} \left (-\frac {x^{4} {\mathrm e}^{2 x}}{2}+{\mathrm e}^{2 x} x^{6}-\frac {x^{8} {\mathrm e}^{2 x}}{2}\right )+x^{2} {\mathrm e}^{2}\)	\(279\)
parts	\(-64 \,{\mathrm e}^{4 x} x^{12}+96 \,{\mathrm e}^{4 x} x^{10}-64 \,{\mathrm e}^{4 x} x^{8}+16 \,{\mathrm e}^{4 x} x^{6}+16 \,{\mathrm e}^{4 x} x^{14}-32 \,{\mathrm e} \left (\frac {x^{3} {\mathrm e}^{4 x}}{4}-\frac {3 x^{2} {\mathrm e}^{4 x}}{16}+\frac {3 x \,{\mathrm e}^{4 x}}{32}-\frac {3 \,{\mathrm e}^{4 x}}{128}\right )-32 \,{\mathrm e} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}-\frac {x^{3} {\mathrm e}^{4 x}}{4}+\frac {3 x^{2} {\mathrm e}^{4 x}}{16}-\frac {3 x \,{\mathrm e}^{4 x}}{32}+\frac {3 \,{\mathrm e}^{4 x}}{128}\right )+16 \,{\mathrm e}^{8 x} x^{6}+64 \,{\mathrm e}^{7 x} x^{6}-64 \,{\mathrm e}^{7 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{6}-192 \,{\mathrm e}^{6 x} x^{8}+96 \,{\mathrm e}^{6 x} x^{10}-64 \,{\mathrm e}^{5 x} x^{12}+192 \,{\mathrm e}^{5 x} x^{10}-192 \,{\mathrm e}^{5 x} x^{8}+64 x^{6} {\mathrm e}^{5 x}+x^{2} {\mathrm e}^{2}+16 \,{\mathrm e} \left (-x^{4} {\mathrm e}^{3 x}+x^{6} {\mathrm e}^{3 x}\right )-16 \,{\mathrm e} \left (\frac {x^{4} {\mathrm e}^{2 x}}{2}+\frac {x^{8} {\mathrm e}^{2 x}}{2}-{\mathrm e}^{2 x} x^{6}\right )\)	\(280\)

Input:

int((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+ 
(576*x^10+960*x^9-1152*x^8-1536*x^7+576*x^6+576*x^5)*exp(x)^6+(-320*x^12-7 
68*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5+((-32 
*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-25 
6*x^8-512*x^7+64*x^6+96*x^5)*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1) 
*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x)^2+2*x 
*exp(1)^2,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

16*exp(8*x)*x^6+(-64*x^8+64*x^6)*exp(7*x)+(96*x^10-192*x^8+96*x^6)*exp(6*x 
)+(-64*x^12+192*x^10-192*x^8+64*x^6)*exp(5*x)+(16*x^14-64*x^12+96*x^10-64* 
x^8+16*x^6-8*x^4*exp(1))*exp(4*x)+(16*x^6-16*x^4)*exp(1+3*x)+(-8*x^8+16*x^ 
6-8*x^4)*exp(1+2*x)+x^2*exp(2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (31) = 62\).

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.09 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx={\left (16 \, x^{6} e^{\left (8 \, x + 4\right )} + x^{2} e^{6} - 64 \, {\left (x^{8} - x^{6}\right )} e^{\left (7 \, x + 4\right )} + 96 \, {\left (x^{10} - 2 \, x^{8} + x^{6}\right )} e^{\left (6 \, x + 4\right )} - 64 \, {\left (x^{12} - 3 \, x^{10} + 3 \, x^{8} - x^{6}\right )} e^{\left (5 \, x + 4\right )} - 8 \, {\left (x^{4} e^{3} - 2 \, {\left (x^{14} - 4 \, x^{12} + 6 \, x^{10} - 4 \, x^{8} + x^{6}\right )} e^{2}\right )} e^{\left (4 \, x + 2\right )} + 16 \, {\left (x^{6} - x^{4}\right )} e^{\left (3 \, x + 5\right )} - 8 \, {\left (x^{8} - 2 \, x^{6} + x^{4}\right )} e^{\left (2 \, x + 5\right )}\right )} e^{\left (-4\right )} \] Input:

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp 
(x)^7+(576*x^10+960*x^9-1152*x^8-1536*x^7+576*x^6+576*x^5)*exp(x)^6+(-320* 
x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5 
+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960* 
x^9-256*x^8-512*x^7+64*x^6+96*x^5)*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)* 
exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x) 
^2+2*x*exp(1)^2,x, algorithm="fricas")
 

Output:

(16*x^6*e^(8*x + 4) + x^2*e^6 - 64*(x^8 - x^6)*e^(7*x + 4) + 96*(x^10 - 2* 
x^8 + x^6)*e^(6*x + 4) - 64*(x^12 - 3*x^10 + 3*x^8 - x^6)*e^(5*x + 4) - 8* 
(x^4*e^3 - 2*(x^14 - 4*x^12 + 6*x^10 - 4*x^8 + x^6)*e^2)*e^(4*x + 2) + 16* 
(x^6 - x^4)*e^(3*x + 5) - 8*(x^8 - 2*x^6 + x^4)*e^(2*x + 5))*e^(-4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (27) = 54\).

Time = 0.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.25 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=16 x^{6} e^{8 x} + x^{2} e^{2} + \left (- 64 x^{8} + 64 x^{6}\right ) e^{7 x} + \left (16 e x^{6} - 16 e x^{4}\right ) e^{3 x} + \left (96 x^{10} - 192 x^{8} + 96 x^{6}\right ) e^{6 x} + \left (- 8 e x^{8} + 16 e x^{6} - 8 e x^{4}\right ) e^{2 x} + \left (- 64 x^{12} + 192 x^{10} - 192 x^{8} + 64 x^{6}\right ) e^{5 x} + \left (16 x^{14} - 64 x^{12} + 96 x^{10} - 64 x^{8} + 16 x^{6} - 8 e x^{4}\right ) e^{4 x} \] Input:

integrate((128*x**6+96*x**5)*exp(x)**8+(-448*x**8-512*x**7+448*x**6+384*x* 
*5)*exp(x)**7+(576*x**10+960*x**9-1152*x**8-1536*x**7+576*x**6+576*x**5)*e 
xp(x)**6+(-320*x**12-768*x**11+960*x**10+1920*x**9-960*x**8-1536*x**7+320* 
x**6+384*x**5)*exp(x)**5+((-32*x**4-32*x**3)*exp(1)+64*x**14+224*x**13-256 
*x**12-768*x**11+384*x**10+960*x**9-256*x**8-512*x**7+64*x**6+96*x**5)*exp 
(x)**4+(48*x**6+96*x**5-48*x**4-64*x**3)*exp(1)*exp(x)**3+(-16*x**8-64*x** 
7+32*x**6+96*x**5-16*x**4-32*x**3)*exp(1)*exp(x)**2+2*x*exp(1)**2,x)
 

Output:

16*x**6*exp(8*x) + x**2*exp(2) + (-64*x**8 + 64*x**6)*exp(7*x) + (16*E*x** 
6 - 16*E*x**4)*exp(3*x) + (96*x**10 - 192*x**8 + 96*x**6)*exp(6*x) + (-8*E 
*x**8 + 16*E*x**6 - 8*E*x**4)*exp(2*x) + (-64*x**12 + 192*x**10 - 192*x**8 
 + 64*x**6)*exp(5*x) + (16*x**14 - 64*x**12 + 96*x**10 - 64*x**8 + 16*x**6 
 - 8*E*x**4)*exp(4*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (31) = 62\).

Time = 0.06 (sec) , antiderivative size = 722, normalized size of antiderivative = 22.56 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp 
(x)^7+(576*x^10+960*x^9-1152*x^8-1536*x^7+576*x^6+576*x^5)*exp(x)^6+(-320* 
x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5 
+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960* 
x^9-256*x^8-512*x^7+64*x^6+96*x^5)*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)* 
exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x) 
^2+2*x*exp(1)^2,x, algorithm="maxima")
 

Output:

16*x^6*e^(8*x) + x^2*e^2 - 64*(x^8 - x^6)*e^(7*x) + 96*(x^10 - 2*x^8 + x^6 
)*e^(6*x) - 64*(x^12 - 3*x^10 + 3*x^8 - x^6)*e^(5*x) + 1/8192*(131072*x^14 
 - 458752*x^13 + 1490944*x^12 - 4472832*x^11 + 12300288*x^10 - 30750720*x^ 
9 + 69189120*x^8 - 138378240*x^7 + 242161920*x^6 - 363242880*x^5 + 4540536 
00*x^4 - 454053600*x^3 + 340540200*x^2 - 170270100*x + 42567525)*e^(4*x) + 
 7/8192*(65536*x^13 - 212992*x^12 + 638976*x^11 - 1757184*x^10 + 4392960*x 
^9 - 9884160*x^8 + 19768320*x^7 - 34594560*x^6 + 51891840*x^5 - 64864800*x 
^4 + 64864800*x^3 - 48648600*x^2 + 24324300*x - 6081075)*e^(4*x) - 1/256*( 
16384*x^12 - 49152*x^11 + 135168*x^10 - 337920*x^9 + 760320*x^8 - 1520640* 
x^7 + 2661120*x^6 - 3991680*x^5 + 4989600*x^4 - 4989600*x^3 + 3742200*x^2 
- 1871100*x + 467775)*e^(4*x) - 3/256*(16384*x^11 - 45056*x^10 + 112640*x^ 
9 - 253440*x^8 + 506880*x^7 - 887040*x^6 + 1330560*x^5 - 1663200*x^4 + 166 
3200*x^3 - 1247400*x^2 + 623700*x - 155925)*e^(4*x) + 3/128*(4096*x^10 - 1 
0240*x^9 + 23040*x^8 - 46080*x^7 + 80640*x^6 - 120960*x^5 + 151200*x^4 - 1 
51200*x^3 + 113400*x^2 - 56700*x + 14175)*e^(4*x) + 15/128*(2048*x^9 - 460 
8*x^8 + 9216*x^7 - 16128*x^6 + 24192*x^5 - 30240*x^4 + 30240*x^3 - 22680*x 
^2 + 11340*x - 2835)*e^(4*x) - 1/8*(512*x^8 - 1024*x^7 + 1792*x^6 - 2688*x 
^5 + 3360*x^4 - 3360*x^3 + 2520*x^2 - 1260*x + 315)*e^(4*x) - 1/8*(1024*x^ 
7 - 1792*x^6 + 2688*x^5 - 3360*x^4 + 3360*x^3 - 2520*x^2 + 1260*x - 315)*e 
^(4*x) + 1/16*(256*x^6 - 384*x^5 + 480*x^4 - 480*x^3 + 360*x^2 - 180*x ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (31) = 62\).

Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.69 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=16 \, x^{6} e^{\left (8 \, x\right )} - 8 \, x^{4} e^{\left (4 \, x + 1\right )} + x^{2} e^{2} - 64 \, {\left (x^{8} - x^{6}\right )} e^{\left (7 \, x\right )} + 96 \, {\left (x^{10} - 2 \, x^{8} + x^{6}\right )} e^{\left (6 \, x\right )} - 64 \, {\left (x^{12} - 3 \, x^{10} + 3 \, x^{8} - x^{6}\right )} e^{\left (5 \, x\right )} + 16 \, {\left (x^{14} - 4 \, x^{12} + 6 \, x^{10} - 4 \, x^{8} + x^{6}\right )} e^{\left (4 \, x\right )} + 16 \, {\left (x^{6} - x^{4}\right )} e^{\left (3 \, x + 1\right )} - 8 \, {\left (x^{8} - 2 \, x^{6} + x^{4}\right )} e^{\left (2 \, x + 1\right )} \] Input:

integrate((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp 
(x)^7+(576*x^10+960*x^9-1152*x^8-1536*x^7+576*x^6+576*x^5)*exp(x)^6+(-320* 
x^12-768*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5 
+((-32*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960* 
x^9-256*x^8-512*x^7+64*x^6+96*x^5)*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)* 
exp(1)*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x) 
^2+2*x*exp(1)^2,x, algorithm="giac")
 

Output:

16*x^6*e^(8*x) - 8*x^4*e^(4*x + 1) + x^2*e^2 - 64*(x^8 - x^6)*e^(7*x) + 96 
*(x^10 - 2*x^8 + x^6)*e^(6*x) - 64*(x^12 - 3*x^10 + 3*x^8 - x^6)*e^(5*x) + 
 16*(x^14 - 4*x^12 + 6*x^10 - 4*x^8 + x^6)*e^(4*x) + 16*(x^6 - x^4)*e^(3*x 
 + 1) - 8*(x^8 - 2*x^6 + x^4)*e^(2*x + 1)
 

Mupad [B] (verification not implemented)

Time = 4.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 6.50 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=16\,x^6\,{\mathrm {e}}^{4\,x}+64\,x^6\,{\mathrm {e}}^{5\,x}+96\,x^6\,{\mathrm {e}}^{6\,x}-64\,x^8\,{\mathrm {e}}^{4\,x}+64\,x^6\,{\mathrm {e}}^{7\,x}-192\,x^8\,{\mathrm {e}}^{5\,x}+16\,x^6\,{\mathrm {e}}^{8\,x}-192\,x^8\,{\mathrm {e}}^{6\,x}+96\,x^{10}\,{\mathrm {e}}^{4\,x}-64\,x^8\,{\mathrm {e}}^{7\,x}+192\,x^{10}\,{\mathrm {e}}^{5\,x}+96\,x^{10}\,{\mathrm {e}}^{6\,x}-64\,x^{12}\,{\mathrm {e}}^{4\,x}-64\,x^{12}\,{\mathrm {e}}^{5\,x}+16\,x^{14}\,{\mathrm {e}}^{4\,x}+x^2\,{\mathrm {e}}^2-8\,x^4\,{\mathrm {e}}^{2\,x+1}-16\,x^4\,{\mathrm {e}}^{3\,x+1}-8\,x^4\,{\mathrm {e}}^{4\,x+1}+16\,x^6\,{\mathrm {e}}^{2\,x+1}+16\,x^6\,{\mathrm {e}}^{3\,x+1}-8\,x^8\,{\mathrm {e}}^{2\,x+1} \] Input:

int(2*x*exp(2) + exp(6*x)*(576*x^5 + 576*x^6 - 1536*x^7 - 1152*x^8 + 960*x 
^9 + 576*x^10) + exp(8*x)*(96*x^5 + 128*x^6) + exp(4*x)*(96*x^5 - exp(1)*( 
32*x^3 + 32*x^4) + 64*x^6 - 512*x^7 - 256*x^8 + 960*x^9 + 384*x^10 - 768*x 
^11 - 256*x^12 + 224*x^13 + 64*x^14) + exp(5*x)*(384*x^5 + 320*x^6 - 1536* 
x^7 - 960*x^8 + 1920*x^9 + 960*x^10 - 768*x^11 - 320*x^12) + exp(7*x)*(384 
*x^5 + 448*x^6 - 512*x^7 - 448*x^8) - exp(3*x)*exp(1)*(64*x^3 + 48*x^4 - 9 
6*x^5 - 48*x^6) - exp(2*x)*exp(1)*(32*x^3 + 16*x^4 - 96*x^5 - 32*x^6 + 64* 
x^7 + 16*x^8),x)
 

Output:

16*x^6*exp(4*x) + 64*x^6*exp(5*x) + 96*x^6*exp(6*x) - 64*x^8*exp(4*x) + 64 
*x^6*exp(7*x) - 192*x^8*exp(5*x) + 16*x^6*exp(8*x) - 192*x^8*exp(6*x) + 96 
*x^10*exp(4*x) - 64*x^8*exp(7*x) + 192*x^10*exp(5*x) + 96*x^10*exp(6*x) - 
64*x^12*exp(4*x) - 64*x^12*exp(5*x) + 16*x^14*exp(4*x) + x^2*exp(2) - 8*x^ 
4*exp(2*x + 1) - 16*x^4*exp(3*x + 1) - 8*x^4*exp(4*x + 1) + 16*x^6*exp(2*x 
 + 1) + 16*x^6*exp(3*x + 1) - 8*x^8*exp(2*x + 1)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 7.00 \[ \int \left (2 e^2 x+e^{1+3 x} \left (-64 x^3-48 x^4+96 x^5+48 x^6\right )+e^{8 x} \left (96 x^5+128 x^6\right )+e^{7 x} \left (384 x^5+448 x^6-512 x^7-448 x^8\right )+e^{1+2 x} \left (-32 x^3-16 x^4+96 x^5+32 x^6-64 x^7-16 x^8\right )+e^{6 x} \left (576 x^5+576 x^6-1536 x^7-1152 x^8+960 x^9+576 x^{10}\right )+e^{5 x} \left (384 x^5+320 x^6-1536 x^7-960 x^8+1920 x^9+960 x^{10}-768 x^{11}-320 x^{12}\right )+e^{4 x} \left (96 x^5+64 x^6-512 x^7-256 x^8+960 x^9+384 x^{10}-768 x^{11}-256 x^{12}+224 x^{13}+64 x^{14}+e \left (-32 x^3-32 x^4\right )\right )\right ) \, dx=x^{2} \left (16 e^{8 x} x^{4}-64 e^{7 x} x^{6}+64 e^{7 x} x^{4}+96 e^{6 x} x^{8}-192 e^{6 x} x^{6}+96 e^{6 x} x^{4}-64 e^{5 x} x^{10}+192 e^{5 x} x^{8}-192 e^{5 x} x^{6}+64 e^{5 x} x^{4}-8 e^{4 x} e \,x^{2}+16 e^{4 x} x^{12}-64 e^{4 x} x^{10}+96 e^{4 x} x^{8}-64 e^{4 x} x^{6}+16 e^{4 x} x^{4}+16 e^{3 x} e \,x^{4}-16 e^{3 x} e \,x^{2}-8 e^{2 x} e \,x^{6}+16 e^{2 x} e \,x^{4}-8 e^{2 x} e \,x^{2}+e^{2}\right ) \] Input:

int((128*x^6+96*x^5)*exp(x)^8+(-448*x^8-512*x^7+448*x^6+384*x^5)*exp(x)^7+ 
(576*x^10+960*x^9-1152*x^8-1536*x^7+576*x^6+576*x^5)*exp(x)^6+(-320*x^12-7 
68*x^11+960*x^10+1920*x^9-960*x^8-1536*x^7+320*x^6+384*x^5)*exp(x)^5+((-32 
*x^4-32*x^3)*exp(1)+64*x^14+224*x^13-256*x^12-768*x^11+384*x^10+960*x^9-25 
6*x^8-512*x^7+64*x^6+96*x^5)*exp(x)^4+(48*x^6+96*x^5-48*x^4-64*x^3)*exp(1) 
*exp(x)^3+(-16*x^8-64*x^7+32*x^6+96*x^5-16*x^4-32*x^3)*exp(1)*exp(x)^2+2*x 
*exp(1)^2,x)
                                                                                    
                                                                                    
 

Output:

x**2*(16*e**(8*x)*x**4 - 64*e**(7*x)*x**6 + 64*e**(7*x)*x**4 + 96*e**(6*x) 
*x**8 - 192*e**(6*x)*x**6 + 96*e**(6*x)*x**4 - 64*e**(5*x)*x**10 + 192*e** 
(5*x)*x**8 - 192*e**(5*x)*x**6 + 64*e**(5*x)*x**4 - 8*e**(4*x)*e*x**2 + 16 
*e**(4*x)*x**12 - 64*e**(4*x)*x**10 + 96*e**(4*x)*x**8 - 64*e**(4*x)*x**6 
+ 16*e**(4*x)*x**4 + 16*e**(3*x)*e*x**4 - 16*e**(3*x)*e*x**2 - 8*e**(2*x)* 
e*x**6 + 16*e**(2*x)*e*x**4 - 8*e**(2*x)*e*x**2 + e**2)