\(\int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} (1700 x+2500 x^2+1500 x^3)+e^{2 x} (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5)+e^x (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7)}{125 x} \, dx\) [226]

Optimal result

Integrand size = 143, antiderivative size = 20 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=\left (\frac {4}{5}+e^x+x+x^2\right )^4-\frac {\log (x)}{5} \] Output:

(x+4/5+x^2+exp(x))^4-1/5*ln(x)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(106\) vs. \(2(20)=40\).

Time = 2.96 (sec) , antiderivative size = 106, normalized size of antiderivative = 5.30 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=\frac {1}{125} \left (125 e^{4 x}+100 e^{3 x} \left (4+5 x+5 x^2\right )+30 e^{2 x} \left (4+5 x+5 x^2\right )^2+4 e^x \left (4+5 x+5 x^2\right )^3+x \left (256+736 x+1360 x^2+1805 x^3+1700 x^4+1150 x^5+500 x^6+125 x^7\right )-25 \log (x)\right ) \] Input:

Integrate[(-25 + 256*x + 500*E^(4*x)*x + 1472*x^2 + 4080*x^3 + 7220*x^4 + 
8500*x^5 + 6900*x^6 + 3500*x^7 + 1000*x^8 + E^(3*x)*(1700*x + 2500*x^2 + 1 
500*x^3) + E^(2*x)*(2160*x + 6300*x^2 + 8400*x^3 + 6000*x^4 + 1500*x^5) + 
E^x*(1216*x + 5280*x^2 + 10860*x^3 + 13700*x^4 + 10200*x^5 + 4500*x^6 + 50 
0*x^7))/(125*x),x]
 

Output:

(125*E^(4*x) + 100*E^(3*x)*(4 + 5*x + 5*x^2) + 30*E^(2*x)*(4 + 5*x + 5*x^2 
)^2 + 4*E^x*(4 + 5*x + 5*x^2)^3 + x*(256 + 736*x + 1360*x^2 + 1805*x^3 + 1 
700*x^4 + 1150*x^5 + 500*x^6 + 125*x^7) - 25*Log[x])/125
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(20)=40\).

Time = 0.80 (sec) , antiderivative size = 175, normalized size of antiderivative = 8.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {27, 25, 2010, 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.

\(\displaystyle \int \frac {1000 x^8+3500 x^7+6900 x^6+8500 x^5+7220 x^4+4080 x^3+1472 x^2+e^{3 x} \left (1500 x^3+2500 x^2+1700 x\right )+e^{2 x} \left (1500 x^5+6000 x^4+8400 x^3+6300 x^2+2160 x\right )+e^x \left (500 x^7+4500 x^6+10200 x^5+13700 x^4+10860 x^3+5280 x^2+1216 x\right )+500 e^{4 x} x+256 x-25}{125 x} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{125} \int -\frac {-1000 x^8-3500 x^7-6900 x^6-8500 x^5-7220 x^4-4080 x^3-1472 x^2-500 e^{4 x} x-256 x-100 e^{3 x} \left (15 x^3+25 x^2+17 x\right )-60 e^{2 x} \left (25 x^5+100 x^4+140 x^3+105 x^2+36 x\right )-4 e^x \left (125 x^7+1125 x^6+2550 x^5+3425 x^4+2715 x^3+1320 x^2+304 x\right )+25}{x}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{125} \int \frac {-1000 x^8-3500 x^7-6900 x^6-8500 x^5-7220 x^4-4080 x^3-1472 x^2-500 e^{4 x} x-256 x-100 e^{3 x} \left (15 x^3+25 x^2+17 x\right )-60 e^{2 x} \left (25 x^5+100 x^4+140 x^3+105 x^2+36 x\right )-4 e^x \left (125 x^7+1125 x^6+2550 x^5+3425 x^4+2715 x^3+1320 x^2+304 x\right )+25}{x}dx\)

\(\Big \downarrow \) 2010

\(\displaystyle -\frac {1}{125} \int \left (-4 e^x \left (5 x^2+35 x+19\right ) \left (5 x^2+5 x+4\right )^2-60 e^{2 x} \left (5 x^2+15 x+9\right ) \left (5 x^2+5 x+4\right )-500 e^{4 x}-100 e^{3 x} \left (15 x^2+25 x+17\right )+\frac {-1000 x^8-3500 x^7-6900 x^6-8500 x^5-7220 x^4-4080 x^3-1472 x^2-256 x+25}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{125} \left (125 x^8+500 x^7+500 e^x x^6+1150 x^6+1500 e^x x^5+1700 x^5+2700 e^x x^4+750 e^{2 x} x^4+1805 x^4+2900 e^x x^3+1500 e^{2 x} x^3+1360 x^3+2160 e^x x^2+1950 e^{2 x} x^2+500 e^{3 x} x^2+736 x^2+960 e^x x+1200 e^{2 x} x+500 e^{3 x} x+256 x+256 e^x+480 e^{2 x}+400 e^{3 x}+125 e^{4 x}-25 \log (x)\right )\)

Input:

Int[(-25 + 256*x + 500*E^(4*x)*x + 1472*x^2 + 4080*x^3 + 7220*x^4 + 8500*x 
^5 + 6900*x^6 + 3500*x^7 + 1000*x^8 + E^(3*x)*(1700*x + 2500*x^2 + 1500*x^ 
3) + E^(2*x)*(2160*x + 6300*x^2 + 8400*x^3 + 6000*x^4 + 1500*x^5) + E^x*(1 
216*x + 5280*x^2 + 10860*x^3 + 13700*x^4 + 10200*x^5 + 4500*x^6 + 500*x^7) 
)/(125*x),x]
 

Output:

(256*E^x + 480*E^(2*x) + 400*E^(3*x) + 125*E^(4*x) + 256*x + 960*E^x*x + 1 
200*E^(2*x)*x + 500*E^(3*x)*x + 736*x^2 + 2160*E^x*x^2 + 1950*E^(2*x)*x^2 
+ 500*E^(3*x)*x^2 + 1360*x^3 + 2900*E^x*x^3 + 1500*E^(2*x)*x^3 + 1805*x^4 
+ 2700*E^x*x^4 + 750*E^(2*x)*x^4 + 1700*x^5 + 1500*E^x*x^5 + 1150*x^6 + 50 
0*E^x*x^6 + 500*x^7 + 125*x^8 - 25*Log[x])/125
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(15)=30\).

Time = 8.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 6.15

method	result	size
risch	\(x^{8}+4 x^{7}+\frac {46 x^{6}}{5}+\frac {68 x^{5}}{5}+\frac {361 x^{4}}{25}+\frac {272 x^{3}}{25}+\frac {736 x^{2}}{125}+\frac {256 x}{125}+\frac {256}{625}-\frac {\ln \left (x \right )}{5}+{\mathrm e}^{4 x}+\frac {\left (500 x^{2}+500 x +400\right ) {\mathrm e}^{3 x}}{125}+\frac {\left (750 x^{4}+1500 x^{3}+1950 x^{2}+1200 x +480\right ) {\mathrm e}^{2 x}}{125}+\frac {\left (500 x^{6}+1500 x^{5}+2700 x^{4}+2900 x^{3}+2160 x^{2}+960 x +256\right ) {\mathrm e}^{x}}{125}\)	\(123\)
default	\(\frac {256 x}{125}+4 x^{2} {\mathrm e}^{3 x}+\frac {78 \,{\mathrm e}^{2 x} x^{2}}{5}+4 x^{6} {\mathrm e}^{x}+\frac {48 x \,{\mathrm e}^{2 x}}{5}+\frac {108 \,{\mathrm e}^{x} x^{4}}{5}+\frac {432 \,{\mathrm e}^{x} x^{2}}{25}+\frac {192 \,{\mathrm e}^{x} x}{25}+4 x \,{\mathrm e}^{3 x}+\frac {116 \,{\mathrm e}^{x} x^{3}}{5}+12 x^{5} {\mathrm e}^{x}+\frac {96 \,{\mathrm e}^{2 x}}{25}+x^{8}+\frac {16 \,{\mathrm e}^{3 x}}{5}+\frac {68 x^{5}}{5}+{\mathrm e}^{4 x}-\frac {\ln \left (x \right )}{5}+4 x^{7}+\frac {46 x^{6}}{5}+\frac {736 x^{2}}{125}+\frac {256 \,{\mathrm e}^{x}}{125}+\frac {272 x^{3}}{25}+\frac {361 x^{4}}{25}+12 \,{\mathrm e}^{2 x} x^{3}+6 \,{\mathrm e}^{2 x} x^{4}\)	\(152\)
parallelrisch	\(\frac {256 x}{125}+4 x^{2} {\mathrm e}^{3 x}+\frac {78 \,{\mathrm e}^{2 x} x^{2}}{5}+4 x^{6} {\mathrm e}^{x}+\frac {48 x \,{\mathrm e}^{2 x}}{5}+\frac {108 \,{\mathrm e}^{x} x^{4}}{5}+\frac {432 \,{\mathrm e}^{x} x^{2}}{25}+\frac {192 \,{\mathrm e}^{x} x}{25}+4 x \,{\mathrm e}^{3 x}+\frac {116 \,{\mathrm e}^{x} x^{3}}{5}+12 x^{5} {\mathrm e}^{x}+\frac {96 \,{\mathrm e}^{2 x}}{25}+x^{8}+\frac {16 \,{\mathrm e}^{3 x}}{5}+\frac {68 x^{5}}{5}+{\mathrm e}^{4 x}-\frac {\ln \left (x \right )}{5}+4 x^{7}+\frac {46 x^{6}}{5}+\frac {736 x^{2}}{125}+\frac {256 \,{\mathrm e}^{x}}{125}+\frac {272 x^{3}}{25}+\frac {361 x^{4}}{25}+12 \,{\mathrm e}^{2 x} x^{3}+6 \,{\mathrm e}^{2 x} x^{4}\)	\(152\)
parts	\(\frac {256 x}{125}+4 x^{2} {\mathrm e}^{3 x}+\frac {78 \,{\mathrm e}^{2 x} x^{2}}{5}+4 x^{6} {\mathrm e}^{x}+\frac {48 x \,{\mathrm e}^{2 x}}{5}+\frac {108 \,{\mathrm e}^{x} x^{4}}{5}+\frac {432 \,{\mathrm e}^{x} x^{2}}{25}+\frac {192 \,{\mathrm e}^{x} x}{25}+4 x \,{\mathrm e}^{3 x}+\frac {116 \,{\mathrm e}^{x} x^{3}}{5}+12 x^{5} {\mathrm e}^{x}+\frac {96 \,{\mathrm e}^{2 x}}{25}+x^{8}+\frac {16 \,{\mathrm e}^{3 x}}{5}+\frac {68 x^{5}}{5}+{\mathrm e}^{4 x}-\frac {\ln \left (x \right )}{5}+4 x^{7}+\frac {46 x^{6}}{5}+\frac {736 x^{2}}{125}+\frac {256 \,{\mathrm e}^{x}}{125}+\frac {272 x^{3}}{25}+\frac {361 x^{4}}{25}+12 \,{\mathrm e}^{2 x} x^{3}+6 \,{\mathrm e}^{2 x} x^{4}\)	\(152\)

Input:

int(1/125*(500*x*exp(x)^4+(1500*x^3+2500*x^2+1700*x)*exp(x)^3+(1500*x^5+60 
00*x^4+8400*x^3+6300*x^2+2160*x)*exp(x)^2+(500*x^7+4500*x^6+10200*x^5+1370 
0*x^4+10860*x^3+5280*x^2+1216*x)*exp(x)+1000*x^8+3500*x^7+6900*x^6+8500*x^ 
5+7220*x^4+4080*x^3+1472*x^2+256*x-25)/x,x,method=_RETURNVERBOSE)
 

Output:

x^8+4*x^7+46/5*x^6+68/5*x^5+361/25*x^4+272/25*x^3+736/125*x^2+256/125*x+25 
6/625-1/5*ln(x)+exp(x)^4+1/125*(500*x^2+500*x+400)*exp(x)^3+1/125*(750*x^4 
+1500*x^3+1950*x^2+1200*x+480)*exp(x)^2+1/125*(500*x^6+1500*x^5+2700*x^4+2 
900*x^3+2160*x^2+960*x+256)*exp(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 6.05 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=x^{8} + 4 \, x^{7} + \frac {46}{5} \, x^{6} + \frac {68}{5} \, x^{5} + \frac {361}{25} \, x^{4} + \frac {272}{25} \, x^{3} + \frac {736}{125} \, x^{2} + \frac {4}{5} \, {\left (5 \, x^{2} + 5 \, x + 4\right )} e^{\left (3 \, x\right )} + \frac {6}{25} \, {\left (25 \, x^{4} + 50 \, x^{3} + 65 \, x^{2} + 40 \, x + 16\right )} e^{\left (2 \, x\right )} + \frac {4}{125} \, {\left (125 \, x^{6} + 375 \, x^{5} + 675 \, x^{4} + 725 \, x^{3} + 540 \, x^{2} + 240 \, x + 64\right )} e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} - \frac {1}{5} \, \log \left (x\right ) \] Input:

integrate(1/125*(500*x*exp(x)^4+(1500*x^3+2500*x^2+1700*x)*exp(x)^3+(1500* 
x^5+6000*x^4+8400*x^3+6300*x^2+2160*x)*exp(x)^2+(500*x^7+4500*x^6+10200*x^ 
5+13700*x^4+10860*x^3+5280*x^2+1216*x)*exp(x)+1000*x^8+3500*x^7+6900*x^6+8 
500*x^5+7220*x^4+4080*x^3+1472*x^2+256*x-25)/x,x, algorithm="fricas")
 

Output:

x^8 + 4*x^7 + 46/5*x^6 + 68/5*x^5 + 361/25*x^4 + 272/25*x^3 + 736/125*x^2 
+ 4/5*(5*x^2 + 5*x + 4)*e^(3*x) + 6/25*(25*x^4 + 50*x^3 + 65*x^2 + 40*x + 
16)*e^(2*x) + 4/125*(125*x^6 + 375*x^5 + 675*x^4 + 725*x^3 + 540*x^2 + 240 
*x + 64)*e^x + 256/125*x + e^(4*x) - 1/5*log(x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (17) = 34\).

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 6.80 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=x^{8} + 4 x^{7} + \frac {46 x^{6}}{5} + \frac {68 x^{5}}{5} + \frac {361 x^{4}}{25} + \frac {272 x^{3}}{25} + \frac {736 x^{2}}{125} + \frac {256 x}{125} + \frac {\left (62500 x^{2} + 62500 x + 50000\right ) e^{3 x}}{15625} + \frac {\left (93750 x^{4} + 187500 x^{3} + 243750 x^{2} + 150000 x + 60000\right ) e^{2 x}}{15625} + \frac {\left (62500 x^{6} + 187500 x^{5} + 337500 x^{4} + 362500 x^{3} + 270000 x^{2} + 120000 x + 32000\right ) e^{x}}{15625} + e^{4 x} - \frac {\log {\left (x \right )}}{5} \] Input:

integrate(1/125*(500*x*exp(x)**4+(1500*x**3+2500*x**2+1700*x)*exp(x)**3+(1 
500*x**5+6000*x**4+8400*x**3+6300*x**2+2160*x)*exp(x)**2+(500*x**7+4500*x* 
*6+10200*x**5+13700*x**4+10860*x**3+5280*x**2+1216*x)*exp(x)+1000*x**8+350 
0*x**7+6900*x**6+8500*x**5+7220*x**4+4080*x**3+1472*x**2+256*x-25)/x,x)
 

Output:

x**8 + 4*x**7 + 46*x**6/5 + 68*x**5/5 + 361*x**4/25 + 272*x**3/25 + 736*x* 
*2/125 + 256*x/125 + (62500*x**2 + 62500*x + 50000)*exp(3*x)/15625 + (9375 
0*x**4 + 187500*x**3 + 243750*x**2 + 150000*x + 60000)*exp(2*x)/15625 + (6 
2500*x**6 + 187500*x**5 + 337500*x**4 + 362500*x**3 + 270000*x**2 + 120000 
*x + 32000)*exp(x)/15625 + exp(4*x) - log(x)/5
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (23) = 46\).

Time = 0.05 (sec) , antiderivative size = 279, normalized size of antiderivative = 13.95 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=x^{8} + 4 \, x^{7} + \frac {46}{5} \, x^{6} + \frac {68}{5} \, x^{5} + \frac {361}{25} \, x^{4} + \frac {272}{25} \, x^{3} + \frac {736}{125} \, x^{2} + \frac {4}{9} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {20}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} + 3 \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + 6 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {84}{5} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {63}{5} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} + 36 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {408}{5} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {548}{5} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {2172}{25} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {1056}{25} \, {\left (x - 1\right )} e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} + \frac {68}{15} \, e^{\left (3 \, x\right )} + \frac {216}{25} \, e^{\left (2 \, x\right )} + \frac {1216}{125} \, e^{x} - \frac {1}{5} \, \log \left (x\right ) \] Input:

integrate(1/125*(500*x*exp(x)^4+(1500*x^3+2500*x^2+1700*x)*exp(x)^3+(1500* 
x^5+6000*x^4+8400*x^3+6300*x^2+2160*x)*exp(x)^2+(500*x^7+4500*x^6+10200*x^ 
5+13700*x^4+10860*x^3+5280*x^2+1216*x)*exp(x)+1000*x^8+3500*x^7+6900*x^6+8 
500*x^5+7220*x^4+4080*x^3+1472*x^2+256*x-25)/x,x, algorithm="maxima")
 

Output:

x^8 + 4*x^7 + 46/5*x^6 + 68/5*x^5 + 361/25*x^4 + 272/25*x^3 + 736/125*x^2 
+ 4/9*(9*x^2 - 6*x + 2)*e^(3*x) + 20/9*(3*x - 1)*e^(3*x) + 3*(2*x^4 - 4*x^ 
3 + 6*x^2 - 6*x + 3)*e^(2*x) + 6*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) + 84/5* 
(2*x^2 - 2*x + 1)*e^(2*x) + 63/5*(2*x - 1)*e^(2*x) + 4*(x^6 - 6*x^5 + 30*x 
^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^x + 36*(x^5 - 5*x^4 + 20*x^3 - 60* 
x^2 + 120*x - 120)*e^x + 408/5*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 54 
8/5*(x^3 - 3*x^2 + 6*x - 6)*e^x + 2172/25*(x^2 - 2*x + 2)*e^x + 1056/25*(x 
 - 1)*e^x + 256/125*x + e^(4*x) + 68/15*e^(3*x) + 216/25*e^(2*x) + 1216/12 
5*e^x - 1/5*log(x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (23) = 46\).

Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 7.55 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=x^{8} + 4 \, x^{7} + 4 \, x^{6} e^{x} + \frac {46}{5} \, x^{6} + 12 \, x^{5} e^{x} + \frac {68}{5} \, x^{5} + 6 \, x^{4} e^{\left (2 \, x\right )} + \frac {108}{5} \, x^{4} e^{x} + \frac {361}{25} \, x^{4} + 12 \, x^{3} e^{\left (2 \, x\right )} + \frac {116}{5} \, x^{3} e^{x} + \frac {272}{25} \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} + \frac {78}{5} \, x^{2} e^{\left (2 \, x\right )} + \frac {432}{25} \, x^{2} e^{x} + \frac {736}{125} \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + \frac {48}{5} \, x e^{\left (2 \, x\right )} + \frac {192}{25} \, x e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} + \frac {16}{5} \, e^{\left (3 \, x\right )} + \frac {96}{25} \, e^{\left (2 \, x\right )} + \frac {256}{125} \, e^{x} - \frac {1}{5} \, \log \left (x\right ) \] Input:

integrate(1/125*(500*x*exp(x)^4+(1500*x^3+2500*x^2+1700*x)*exp(x)^3+(1500* 
x^5+6000*x^4+8400*x^3+6300*x^2+2160*x)*exp(x)^2+(500*x^7+4500*x^6+10200*x^ 
5+13700*x^4+10860*x^3+5280*x^2+1216*x)*exp(x)+1000*x^8+3500*x^7+6900*x^6+8 
500*x^5+7220*x^4+4080*x^3+1472*x^2+256*x-25)/x,x, algorithm="giac")
 

Output:

x^8 + 4*x^7 + 4*x^6*e^x + 46/5*x^6 + 12*x^5*e^x + 68/5*x^5 + 6*x^4*e^(2*x) 
 + 108/5*x^4*e^x + 361/25*x^4 + 12*x^3*e^(2*x) + 116/5*x^3*e^x + 272/25*x^ 
3 + 4*x^2*e^(3*x) + 78/5*x^2*e^(2*x) + 432/25*x^2*e^x + 736/125*x^2 + 4*x* 
e^(3*x) + 48/5*x*e^(2*x) + 192/25*x*e^x + 256/125*x + e^(4*x) + 16/5*e^(3* 
x) + 96/25*e^(2*x) + 256/125*e^x - 1/5*log(x)
 

Mupad [B] (verification not implemented)

Time = 3.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 7.55 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=\frac {256\,x}{125}+\frac {96\,{\mathrm {e}}^{2\,x}}{25}+\frac {16\,{\mathrm {e}}^{3\,x}}{5}+{\mathrm {e}}^{4\,x}+\frac {256\,{\mathrm {e}}^x}{125}-\frac {\ln \left (x\right )}{5}+\frac {48\,x\,{\mathrm {e}}^{2\,x}}{5}+4\,x\,{\mathrm {e}}^{3\,x}+\frac {432\,x^2\,{\mathrm {e}}^x}{25}+\frac {116\,x^3\,{\mathrm {e}}^x}{5}+\frac {108\,x^4\,{\mathrm {e}}^x}{5}+12\,x^5\,{\mathrm {e}}^x+4\,x^6\,{\mathrm {e}}^x+\frac {78\,x^2\,{\mathrm {e}}^{2\,x}}{5}+4\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}+6\,x^4\,{\mathrm {e}}^{2\,x}+\frac {192\,x\,{\mathrm {e}}^x}{25}+\frac {736\,x^2}{125}+\frac {272\,x^3}{25}+\frac {361\,x^4}{25}+\frac {68\,x^5}{5}+\frac {46\,x^6}{5}+4\,x^7+x^8 \] Input:

int(((256*x)/125 + 4*x*exp(4*x) + (exp(3*x)*(1700*x + 2500*x^2 + 1500*x^3) 
)/125 + (exp(2*x)*(2160*x + 6300*x^2 + 8400*x^3 + 6000*x^4 + 1500*x^5))/12 
5 + (exp(x)*(1216*x + 5280*x^2 + 10860*x^3 + 13700*x^4 + 10200*x^5 + 4500* 
x^6 + 500*x^7))/125 + (1472*x^2)/125 + (816*x^3)/25 + (1444*x^4)/25 + 68*x 
^5 + (276*x^6)/5 + 28*x^7 + 8*x^8 - 1/5)/x,x)
 

Output:

(256*x)/125 + (96*exp(2*x))/25 + (16*exp(3*x))/5 + exp(4*x) + (256*exp(x)) 
/125 - log(x)/5 + (48*x*exp(2*x))/5 + 4*x*exp(3*x) + (432*x^2*exp(x))/25 + 
 (116*x^3*exp(x))/5 + (108*x^4*exp(x))/5 + 12*x^5*exp(x) + 4*x^6*exp(x) + 
(78*x^2*exp(2*x))/5 + 4*x^2*exp(3*x) + 12*x^3*exp(2*x) + 6*x^4*exp(2*x) + 
(192*x*exp(x))/25 + (736*x^2)/125 + (272*x^3)/25 + (361*x^4)/25 + (68*x^5) 
/5 + (46*x^6)/5 + 4*x^7 + x^8
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 8.35 \[ \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{125 x} \, dx=\frac {256 x}{125}+12 e^{2 x} x^{3}+\frac {78 e^{2 x} x^{2}}{5}+\frac {108 e^{x} x^{4}}{5}+\frac {116 e^{x} x^{3}}{5}+12 e^{x} x^{5}+\frac {736 x^{2}}{125}+\frac {68 x^{5}}{5}+4 x^{7}+4 e^{x} x^{6}+4 e^{3 x} x +\frac {272 x^{3}}{25}+6 e^{2 x} x^{4}+\frac {432 e^{x} x^{2}}{25}+x^{8}+\frac {46 x^{6}}{5}-\frac {\mathrm {log}\left (x \right )}{5}+\frac {48 e^{2 x} x}{5}+\frac {256 e^{x}}{125}+\frac {96 e^{2 x}}{25}+4 e^{3 x} x^{2}+\frac {361 x^{4}}{25}+e^{4 x}+\frac {16 e^{3 x}}{5}+\frac {192 e^{x} x}{25} \] Input:

int(1/125*(500*x*exp(x)^4+(1500*x^3+2500*x^2+1700*x)*exp(x)^3+(1500*x^5+60 
00*x^4+8400*x^3+6300*x^2+2160*x)*exp(x)^2+(500*x^7+4500*x^6+10200*x^5+1370 
0*x^4+10860*x^3+5280*x^2+1216*x)*exp(x)+1000*x^8+3500*x^7+6900*x^6+8500*x^ 
5+7220*x^4+4080*x^3+1472*x^2+256*x-25)/x,x)
 

Output:

(125*e**(4*x) + 500*e**(3*x)*x**2 + 500*e**(3*x)*x + 400*e**(3*x) + 750*e* 
*(2*x)*x**4 + 1500*e**(2*x)*x**3 + 1950*e**(2*x)*x**2 + 1200*e**(2*x)*x + 
480*e**(2*x) + 500*e**x*x**6 + 1500*e**x*x**5 + 2700*e**x*x**4 + 2900*e**x 
*x**3 + 2160*e**x*x**2 + 960*e**x*x + 256*e**x - 25*log(x) + 125*x**8 + 50 
0*x**7 + 1150*x**6 + 1700*x**5 + 1805*x**4 + 1360*x**3 + 736*x**2 + 256*x) 
/125