\(\int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 (63+42 x+7 x^2)+e^{2 x} (-2500 x^2+1000 x^3-100 x^4)+e^x (-10500 x+700 x^2+980 x^3-140 x^4+e^5 (-150-90 x-10 x^2+10 x^3))}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} (2500 x^2-1000 x^3+100 x^4)+e^x (10500 x-700 x^2-980 x^3+140 x^4)} \, dx\) [2124]

Optimal result

Integrand size = 166, antiderivative size = 30 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=-x+\frac {e^5}{(5-x) \left (7+\frac {10 e^x x}{3+x}\right )} \] Output:

exp(5)/(5-x)/(10*exp(x)*x/(3+x)+7)-x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=-x-\frac {e^5 (3+x)}{(-5+x) \left (21+7 x+10 e^x x\right )} \] Input:

Integrate[(-11025 - 2940*x + 1274*x^2 + 196*x^3 - 49*x^4 + E^5*(63 + 42*x 
+ 7*x^2) + E^(2*x)*(-2500*x^2 + 1000*x^3 - 100*x^4) + E^x*(-10500*x + 700* 
x^2 + 980*x^3 - 140*x^4 + E^5*(-150 - 90*x - 10*x^2 + 10*x^3)))/(11025 + 2 
940*x - 1274*x^2 - 196*x^3 + 49*x^4 + E^(2*x)*(2500*x^2 - 1000*x^3 + 100*x 
^4) + E^x*(10500*x - 700*x^2 - 980*x^3 + 140*x^4)),x]
 

Output:

-x - (E^5*(3 + x))/((-5 + x)*(21 + 7*x + 10*E^x*x))
 

Rubi [F]

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.

\(\displaystyle \int \frac {-49 x^4+196 x^3+1274 x^2+e^5 \left (7 x^2+42 x+63\right )+e^{2 x} \left (-100 x^4+1000 x^3-2500 x^2\right )+e^x \left (-140 x^4+980 x^3+700 x^2+e^5 \left (10 x^3-10 x^2-90 x-150\right )-10500 x\right )-2940 x-11025}{49 x^4-196 x^3-1274 x^2+e^{2 x} \left (100 x^4-1000 x^3+2500 x^2\right )+e^x \left (140 x^4-980 x^3-700 x^2+10500 x\right )+2940 x+11025} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-100 e^{2 x} x^2 (x-5)^2-49 \left (x^2-2 x-15\right )^2+10 e^{x+5} \left (x^3-x^2-9 x-15\right )-140 e^x x (x+3) (x-5)^2+7 e^5 (x+3)^2}{(5-x)^2 \left (\left (10 e^x+7\right ) x+21\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^5 \left (x^3-x^2-9 x-15\right )}{(x-5)^2 x \left (10 e^x x+7 x+21\right )}-\frac {7 e^5 \left (x^3+6 x^2+12 x+9\right )}{(x-5) x \left (10 e^x x+7 x+21\right )^2}-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -77 e^5 \int \frac {1}{\left (10 e^x x+7 x+21\right )^2}dx-\frac {2408}{5} e^5 \int \frac {1}{(x-5) \left (10 e^x x+7 x+21\right )^2}dx+\frac {63}{5} e^5 \int \frac {1}{x \left (10 e^x x+7 x+21\right )^2}dx-7 e^5 \int \frac {x}{\left (10 e^x x+7 x+21\right )^2}dx+e^5 \int \frac {1}{10 e^x x+7 x+21}dx+8 e^5 \int \frac {1}{(x-5)^2 \left (10 e^x x+7 x+21\right )}dx+\frac {48}{5} e^5 \int \frac {1}{(x-5) \left (10 e^x x+7 x+21\right )}dx-\frac {3}{5} e^5 \int \frac {1}{x \left (10 e^x x+7 x+21\right )}dx-x\)

Input:

Int[(-11025 - 2940*x + 1274*x^2 + 196*x^3 - 49*x^4 + E^5*(63 + 42*x + 7*x^ 
2) + E^(2*x)*(-2500*x^2 + 1000*x^3 - 100*x^4) + E^x*(-10500*x + 700*x^2 + 
980*x^3 - 140*x^4 + E^5*(-150 - 90*x - 10*x^2 + 10*x^3)))/(11025 + 2940*x 
- 1274*x^2 - 196*x^3 + 49*x^4 + E^(2*x)*(2500*x^2 - 1000*x^3 + 100*x^4) + 
E^x*(10500*x - 700*x^2 - 980*x^3 + 140*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97

method	result	size
risch	\(-x -\frac {{\mathrm e}^{5} \left (3+x \right )}{\left (-5+x \right ) \left (10 \,{\mathrm e}^{x} x +7 x +21\right )}\)	\(29\)
norman	\(\frac {-21 x^{2}+\left (175-{\mathrm e}^{5}\right ) x +250 \,{\mathrm e}^{x} x -7 x^{3}-10 \,{\mathrm e}^{x} x^{3}+525-3 \,{\mathrm e}^{5}}{10 \,{\mathrm e}^{x} x^{2}-50 \,{\mathrm e}^{x} x +7 x^{2}-14 x -105}\)	\(62\)
parallelrisch	\(-\frac {70 \,{\mathrm e}^{x} x^{3}-1470+49 x^{3}-210 \,{\mathrm e}^{x} x^{2}+7 x \,{\mathrm e}^{5}-700 \,{\mathrm e}^{x} x +21 \,{\mathrm e}^{5}-931 x}{7 \left (10 \,{\mathrm e}^{x} x^{2}-50 \,{\mathrm e}^{x} x +7 x^{2}-14 x -105\right )}\)	\(65\)

Input:

int(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5 
)-140*x^4+980*x^3+700*x^2-10500*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+19 
6*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+(140*x^ 
4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x, 
method=_RETURNVERBOSE)
 

Output:

-x-exp(5)*(3+x)/(-5+x)/(10*exp(x)*x+7*x+21)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=-\frac {7 \, x^{3} - 14 \, x^{2} + {\left (x + 3\right )} e^{5} + 10 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{x} - 105 \, x}{7 \, x^{2} + 10 \, {\left (x^{2} - 5 \, x\right )} e^{x} - 14 \, x - 105} \] Input:

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150) 
*exp(5)-140*x^4+980*x^3+700*x^2-10500*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49* 
x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+( 
140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+110 
25),x, algorithm="fricas")
 

Output:

-(7*x^3 - 14*x^2 + (x + 3)*e^5 + 10*(x^3 - 5*x^2)*e^x - 105*x)/(7*x^2 + 10 
*(x^2 - 5*x)*e^x - 14*x - 105)
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=- x + \frac {- x e^{5} - 3 e^{5}}{7 x^{2} - 14 x + \left (10 x^{2} - 50 x\right ) e^{x} - 105} \] Input:

integrate(((-100*x**4+1000*x**3-2500*x**2)*exp(x)**2+((10*x**3-10*x**2-90* 
x-150)*exp(5)-140*x**4+980*x**3+700*x**2-10500*x)*exp(x)+(7*x**2+42*x+63)* 
exp(5)-49*x**4+196*x**3+1274*x**2-2940*x-11025)/((100*x**4-1000*x**3+2500* 
x**2)*exp(x)**2+(140*x**4-980*x**3-700*x**2+10500*x)*exp(x)+49*x**4-196*x* 
*3-1274*x**2+2940*x+11025),x)
 

Output:

-x + (-x*exp(5) - 3*exp(5))/(7*x**2 - 14*x + (10*x**2 - 50*x)*exp(x) - 105 
)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=-\frac {7 \, x^{3} - 14 \, x^{2} + x {\left (e^{5} - 105\right )} + 10 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{x} + 3 \, e^{5}}{7 \, x^{2} + 10 \, {\left (x^{2} - 5 \, x\right )} e^{x} - 14 \, x - 105} \] Input:

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150) 
*exp(5)-140*x^4+980*x^3+700*x^2-10500*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49* 
x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+( 
140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+110 
25),x, algorithm="maxima")
 

Output:

-(7*x^3 - 14*x^2 + x*(e^5 - 105) + 10*(x^3 - 5*x^2)*e^x + 3*e^5)/(7*x^2 + 
10*(x^2 - 5*x)*e^x - 14*x - 105)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=-\frac {10 \, x^{3} e^{x} + 7 \, x^{3} - 50 \, x^{2} e^{x} - 14 \, x^{2} + x e^{5} - 105 \, x + 3 \, e^{5}}{10 \, x^{2} e^{x} + 7 \, x^{2} - 50 \, x e^{x} - 14 \, x - 105} \] Input:

integrate(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150) 
*exp(5)-140*x^4+980*x^3+700*x^2-10500*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49* 
x^4+196*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+( 
140*x^4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+110 
25),x, algorithm="giac")
 

Output:

-(10*x^3*e^x + 7*x^3 - 50*x^2*e^x - 14*x^2 + x*e^5 - 105*x + 3*e^5)/(10*x^ 
2*e^x + 7*x^2 - 50*x*e^x - 14*x - 105)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=\int -\frac {2940\,x-{\mathrm {e}}^5\,\left (7\,x^2+42\,x+63\right )+{\mathrm {e}}^x\,\left (10500\,x+{\mathrm {e}}^5\,\left (-10\,x^3+10\,x^2+90\,x+150\right )-700\,x^2-980\,x^3+140\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (100\,x^4-1000\,x^3+2500\,x^2\right )-1274\,x^2-196\,x^3+49\,x^4+11025}{2940\,x+{\mathrm {e}}^x\,\left (140\,x^4-980\,x^3-700\,x^2+10500\,x\right )+{\mathrm {e}}^{2\,x}\,\left (100\,x^4-1000\,x^3+2500\,x^2\right )-1274\,x^2-196\,x^3+49\,x^4+11025} \,d x \] Input:

int(-(2940*x - exp(5)*(42*x + 7*x^2 + 63) + exp(x)*(10500*x + exp(5)*(90*x 
 + 10*x^2 - 10*x^3 + 150) - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500* 
x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025)/(2940*x + 
 exp(x)*(10500*x - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 100 
0*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025),x)
 

Output:

int(-(2940*x - exp(5)*(42*x + 7*x^2 + 63) + exp(x)*(10500*x + exp(5)*(90*x 
 + 10*x^2 - 10*x^3 + 150) - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500* 
x^2 - 1000*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025)/(2940*x + 
 exp(x)*(10500*x - 700*x^2 - 980*x^3 + 140*x^4) + exp(2*x)*(2500*x^2 - 100 
0*x^3 + 100*x^4) - 1274*x^2 - 196*x^3 + 49*x^4 + 11025), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {-11025-2940 x+1274 x^2+196 x^3-49 x^4+e^5 \left (63+42 x+7 x^2\right )+e^{2 x} \left (-2500 x^2+1000 x^3-100 x^4\right )+e^x \left (-10500 x+700 x^2+980 x^3-140 x^4+e^5 \left (-150-90 x-10 x^2+10 x^3\right )\right )}{11025+2940 x-1274 x^2-196 x^3+49 x^4+e^{2 x} \left (2500 x^2-1000 x^3+100 x^4\right )+e^x \left (10500 x-700 x^2-980 x^3+140 x^4\right )} \, dx=\frac {-10 e^{x} x^{3}+50 e^{x} x^{2}-e^{5} x -3 e^{5}-7 x^{3}+14 x^{2}+105 x}{10 e^{x} x^{2}-50 e^{x} x +7 x^{2}-14 x -105} \] Input:

int(((-100*x^4+1000*x^3-2500*x^2)*exp(x)^2+((10*x^3-10*x^2-90*x-150)*exp(5 
)-140*x^4+980*x^3+700*x^2-10500*x)*exp(x)+(7*x^2+42*x+63)*exp(5)-49*x^4+19 
6*x^3+1274*x^2-2940*x-11025)/((100*x^4-1000*x^3+2500*x^2)*exp(x)^2+(140*x^ 
4-980*x^3-700*x^2+10500*x)*exp(x)+49*x^4-196*x^3-1274*x^2+2940*x+11025),x)
 

Output:

( - 10*e**x*x**3 + 50*e**x*x**2 - e**5*x - 3*e**5 - 7*x**3 + 14*x**2 + 105 
*x)/(10*e**x*x**2 - 50*e**x*x + 7*x**2 - 14*x - 105)