\(\int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} (-625+500 x-150 x^2+20 x^3-x^4)+e^{3 x} (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5)+e^{2 x} (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6)+e^x (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7)}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} (625-500 x+150 x^2-20 x^3+x^4)+e^{3 x} (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5)+e^{2 x} (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6)+e^x (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7)} \, dx\) [260]

Optimal result

Integrand size = 333, antiderivative size = 26 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-79-x+\frac {x}{5+\left (e^x-x\right )^2 (-5+x)^2} \]

[Out]

-79+1/(5+(exp(x)-x)^2*(-5+x)^2)*x-x

Rubi [F]

\[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=\int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx \]

[In]

Int[(-20 - 275*x^2 + 120*x^3 - 638*x^4 + 500*x^5 - 150*x^6 + 20*x^7 - x^8 + E^(4*x)*(-625 + 500*x - 150*x^2 +
20*x^3 - x^4) + E^(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) + E^(2*x)*(-225 + 50*x - 3741*x^2 + 299
8*x^3 - 900*x^4 + 120*x^5 - 6*x^6) + E^x*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7))/(
25 + 250*x^2 - 100*x^3 + 635*x^4 - 500*x^5 + 150*x^6 - 20*x^7 + x^8 + E^(4*x)*(625 - 500*x + 150*x^2 - 20*x^3
+ x^4) + E^(3*x)*(-2500*x + 2000*x^2 - 600*x^3 + 80*x^4 - 4*x^5) + E^(2*x)*(250 - 100*x + 3760*x^2 - 3000*x^3
+ 900*x^4 - 120*x^5 + 6*x^6) + E^x*(-500*x + 200*x^2 - 2520*x^3 + 2000*x^4 - 600*x^5 + 80*x^6 - 4*x^7)),x]

[Out]

-x + 10*Defer[Int][(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^(-2), x] + 50*Defer[I
nt][1/((-5 + x)*(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2), x] + 10*Defer[Int][x
/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x] + 50*Defer[Int][(E^x*x)/(5 + E^(2
*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x] - 50*Defer[Int][x^2/(5 + E^(2*x)*(-5 + x)^2
- 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x] - 70*Defer[Int][(E^x*x^2)/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x
*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x] + 70*Defer[Int][x^3/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x
+ 25*x^2 - 10*x^3 + x^4)^2, x] + 22*Defer[Int][(E^x*x^3)/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2
- 10*x^3 + x^4)^2, x] - 22*Defer[Int][x^4/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^
4)^2, x] - 2*Defer[Int][(E^x*x^4)/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x]
+ 2*Defer[Int][x^5/(5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^2, x] - Defer[Int][(5
+ E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^(-1), x] - 10*Defer[Int][1/((-5 + x)*(5 +
E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)), x] - 2*Defer[Int][x/(5 + E^(2*x)*(-5 + x)^2
- 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-20-e^{4 x} (-5+x)^4+4 e^{3 x} (-5+x)^4 x-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+2 e^x x \left (250-85 x+1252 x^2-999 x^3+300 x^4-40 x^5+2 x^6\right )}{\left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )^2} \, dx \\ & = \int \left (-1-\frac {5-9 x+2 x^2}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )}+\frac {2 x \left (-20-125 e^x+130 x+200 e^x x-200 x^2-90 e^x x^2+90 x^3+16 e^x x^3-16 x^4-e^x x^4+x^5\right )}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}\right ) \, dx \\ & = -x+2 \int \frac {x \left (-20-125 e^x+130 x+200 e^x x-200 x^2-90 e^x x^2+90 x^3+16 e^x x^3-16 x^4-e^x x^4+x^5\right )}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2} \, dx-\int \frac {5-9 x+2 x^2}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )} \, dx \\ & = -x+2 \int \frac {x \left (20+e^x (-5+x)^3 (-1+x)-130 x+200 x^2-90 x^3+16 x^4-x^5\right )}{(5-x) \left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )^2} \, dx-\int \frac {-5+9 x-2 x^2}{(5-x) \left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )} \, dx \\ & = -x+2 \int \left (-\frac {20+125 e^x-130 x-200 e^x x+200 x^2+90 e^x x^2-90 x^3-16 e^x x^3+16 x^4+e^x x^4-x^5}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {5 \left (-20-125 e^x+130 x+200 e^x x-200 x^2-90 e^x x^2+90 x^3+16 e^x x^3-16 x^4-e^x x^4+x^5\right )}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}\right ) \, dx-\int \left (\frac {1}{5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4}+\frac {10}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )}+\frac {2 x}{5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4}\right ) \, dx \\ & = -x-2 \int \frac {x}{5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4} \, dx-2 \int \frac {20+125 e^x-130 x-200 e^x x+200 x^2+90 e^x x^2-90 x^3-16 e^x x^3+16 x^4+e^x x^4-x^5}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2} \, dx-10 \int \frac {1}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )} \, dx+10 \int \frac {-20-125 e^x+130 x+200 e^x x-200 x^2-90 e^x x^2+90 x^3+16 e^x x^3-16 x^4-e^x x^4+x^5}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2} \, dx-\int \frac {1}{5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4} \, dx \\ & = -x-2 \int \frac {x}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4} \, dx-2 \int \frac {20+e^x (-5+x)^3 (-1+x)-130 x+200 x^2-90 x^3+16 x^4-x^5}{\left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )^2} \, dx-10 \int \frac {1}{(-5+x) \left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )} \, dx+10 \int \frac {20+e^x (-5+x)^3 (-1+x)-130 x+200 x^2-90 x^3+16 x^4-x^5}{(5-x) \left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )^2} \, dx-\int \frac {1}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4} \, dx \\ & = -x-2 \int \frac {x}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4} \, dx-2 \int \left (\frac {20}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {125 e^x}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {130 x}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {200 e^x x}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {200 x^2}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {90 e^x x^2}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {90 x^3}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {16 e^x x^3}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {16 x^4}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {e^x x^4}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {x^5}{\left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}\right ) \, dx-10 \int \frac {1}{(-5+x) \left (5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4\right )} \, dx+10 \int \left (-\frac {20}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {125 e^x}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {130 x}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {200 e^x x}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {200 x^2}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {90 e^x x^2}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {90 x^3}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {16 e^x x^3}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {16 x^4}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}-\frac {e^x x^4}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}+\frac {x^5}{(-5+x) \left (5+25 e^{2 x}-50 e^x x-10 e^{2 x} x+25 x^2+20 e^x x^2+e^{2 x} x^2-10 x^3-2 e^x x^3+x^4\right )^2}\right ) \, dx-\int \frac {1}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.15 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=x \left (-1+\frac {1}{5+e^{2 x} (-5+x)^2-2 e^x (-5+x)^2 x+25 x^2-10 x^3+x^4}\right ) \]

[In]

Integrate[(-20 - 275*x^2 + 120*x^3 - 638*x^4 + 500*x^5 - 150*x^6 + 20*x^7 - x^8 + E^(4*x)*(-625 + 500*x - 150*
x^2 + 20*x^3 - x^4) + E^(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5) + E^(2*x)*(-225 + 50*x - 3741*x^2
+ 2998*x^3 - 900*x^4 + 120*x^5 - 6*x^6) + E^x*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x
^7))/(25 + 250*x^2 - 100*x^3 + 635*x^4 - 500*x^5 + 150*x^6 - 20*x^7 + x^8 + E^(4*x)*(625 - 500*x + 150*x^2 - 2
0*x^3 + x^4) + E^(3*x)*(-2500*x + 2000*x^2 - 600*x^3 + 80*x^4 - 4*x^5) + E^(2*x)*(250 - 100*x + 3760*x^2 - 300
0*x^3 + 900*x^4 - 120*x^5 + 6*x^6) + E^x*(-500*x + 200*x^2 - 2520*x^3 + 2000*x^4 - 600*x^5 + 80*x^6 - 4*x^7)),
x]

[Out]

x*(-1 + (5 + E^(2*x)*(-5 + x)^2 - 2*E^x*(-5 + x)^2*x + 25*x^2 - 10*x^3 + x^4)^(-1))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(25)=50\).

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46

method	result	size
risch	\(-x +\frac {x}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\)	\(64\)
parallelrisch	\(-\frac {x^{5}-2 \,{\mathrm e}^{x} x^{4}+50+{\mathrm e}^{2 x} x^{3}-75 x^{3}+150 \,{\mathrm e}^{x} x^{2}-75 x \,{\mathrm e}^{2 x}+250 x^{2}-500 \,{\mathrm e}^{x} x +250 \,{\mathrm e}^{2 x}+4 x}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\)	\(118\)
norman	\(\frac {-250 x^{2}-250 \,{\mathrm e}^{2 x}+75 x^{3}-150 \,{\mathrm e}^{x} x^{2}+75 x \,{\mathrm e}^{2 x}+500 \,{\mathrm e}^{x} x -4 x -x^{5}+2 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{3}-50}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} x^{3}+x^{4}-10 x \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{2}-10 x^{3}+25 \,{\mathrm e}^{2 x}-50 \,{\mathrm e}^{x} x +25 x^{2}+5}\)	\(120\)

[In]

int(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3-2000*x^2+2500*x)*exp(x)^3+(-6*x^6+120*x^5-
900*x^4+2998*x^3-3741*x^2+50*x-225)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(x)-x^8
+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^4+(-4*x^5+80*x^4-60
0*x^3+2000*x^2-2500*x)*exp(x)^3+(6*x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*x^6-60
0*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5+635*x^4-100*x^3+250*x^2+25),x,method=
_RETURNVERBOSE)

[Out]

-x+x/(exp(x)^2*x^2-2*exp(x)*x^3+x^4-10*x*exp(x)^2+20*exp(x)*x^2-10*x^3+25*exp(x)^2-50*exp(x)*x+25*x^2+5)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 10 \, x^{4} + 25 \, x^{3} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )} e^{x} + 4 \, x}{x^{4} - 10 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{x} + 5} \]

[In]

integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3-2000*x^2+2500*x)*exp(x)^3+(-6*x^6+12
0*x^5-900*x^4+2998*x^3-3741*x^2+50*x-225)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(
x)-x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^4+(-4*x^5+80*
x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6*x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*
x^6-600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5+635*x^4-100*x^3+250*x^2+25),x,
algorithm="fricas")

[Out]

-(x^5 - 10*x^4 + 25*x^3 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 2*(x^4 - 10*x^3 + 25*x^2)*e^x + 4*x)/(x^4 - 10*x^3 +
25*x^2 + (x^2 - 10*x + 25)*e^(2*x) - 2*(x^3 - 10*x^2 + 25*x)*e^x + 5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (17) = 34\).

Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=- x + \frac {x}{x^{4} - 10 x^{3} + 25 x^{2} + \left (x^{2} - 10 x + 25\right ) e^{2 x} + \left (- 2 x^{3} + 20 x^{2} - 50 x\right ) e^{x} + 5} \]

[In]

integrate(((-x**4+20*x**3-150*x**2+500*x-625)*exp(x)**4+(4*x**5-80*x**4+600*x**3-2000*x**2+2500*x)*exp(x)**3+(
-6*x**6+120*x**5-900*x**4+2998*x**3-3741*x**2+50*x-225)*exp(x)**2+(4*x**7-80*x**6+600*x**5-1998*x**4+2504*x**3
-170*x**2+500*x)*exp(x)-x**8+20*x**7-150*x**6+500*x**5-638*x**4+120*x**3-275*x**2-20)/((x**4-20*x**3+150*x**2-
500*x+625)*exp(x)**4+(-4*x**5+80*x**4-600*x**3+2000*x**2-2500*x)*exp(x)**3+(6*x**6-120*x**5+900*x**4-3000*x**3
+3760*x**2-100*x+250)*exp(x)**2+(-4*x**7+80*x**6-600*x**5+2000*x**4-2520*x**3+200*x**2-500*x)*exp(x)+x**8-20*x
**7+150*x**6-500*x**5+635*x**4-100*x**3+250*x**2+25),x)

[Out]

-x + x/(x**4 - 10*x**3 + 25*x**2 + (x**2 - 10*x + 25)*exp(2*x) + (-2*x**3 + 20*x**2 - 50*x)*exp(x) + 5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (25) = 50\).

Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 10 \, x^{4} + 25 \, x^{3} + {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )} e^{x} + 4 \, x}{x^{4} - 10 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{x} + 5} \]

[In]

integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3-2000*x^2+2500*x)*exp(x)^3+(-6*x^6+12
0*x^5-900*x^4+2998*x^3-3741*x^2+50*x-225)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(
x)-x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^4+(-4*x^5+80*
x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6*x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*
x^6-600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5+635*x^4-100*x^3+250*x^2+25),x,
algorithm="maxima")

[Out]

-(x^5 - 10*x^4 + 25*x^3 + (x^3 - 10*x^2 + 25*x)*e^(2*x) - 2*(x^4 - 10*x^3 + 25*x^2)*e^x + 4*x)/(x^4 - 10*x^3 +
25*x^2 + (x^2 - 10*x + 25)*e^(2*x) - 2*(x^3 - 10*x^2 + 25*x)*e^x + 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (25) = 50\).

Time = 0.56 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.65 \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=-\frac {x^{5} - 2 \, x^{4} e^{x} - 10 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 20 \, x^{3} e^{x} + 25 \, x^{3} - 10 \, x^{2} e^{\left (2 \, x\right )} - 50 \, x^{2} e^{x} + 25 \, x e^{\left (2 \, x\right )} + 3 \, x}{x^{4} - 2 \, x^{3} e^{x} - 10 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 20 \, x^{2} e^{x} + 25 \, x^{2} - 10 \, x e^{\left (2 \, x\right )} - 50 \, x e^{x} + 25 \, e^{\left (2 \, x\right )} + 5} \]

[In]

integrate(((-x^4+20*x^3-150*x^2+500*x-625)*exp(x)^4+(4*x^5-80*x^4+600*x^3-2000*x^2+2500*x)*exp(x)^3+(-6*x^6+12
0*x^5-900*x^4+2998*x^3-3741*x^2+50*x-225)*exp(x)^2+(4*x^7-80*x^6+600*x^5-1998*x^4+2504*x^3-170*x^2+500*x)*exp(
x)-x^8+20*x^7-150*x^6+500*x^5-638*x^4+120*x^3-275*x^2-20)/((x^4-20*x^3+150*x^2-500*x+625)*exp(x)^4+(-4*x^5+80*
x^4-600*x^3+2000*x^2-2500*x)*exp(x)^3+(6*x^6-120*x^5+900*x^4-3000*x^3+3760*x^2-100*x+250)*exp(x)^2+(-4*x^7+80*
x^6-600*x^5+2000*x^4-2520*x^3+200*x^2-500*x)*exp(x)+x^8-20*x^7+150*x^6-500*x^5+635*x^4-100*x^3+250*x^2+25),x,
algorithm="giac")

[Out]

-(x^5 - 2*x^4*e^x - 10*x^4 + x^3*e^(2*x) + 20*x^3*e^x + 25*x^3 - 10*x^2*e^(2*x) - 50*x^2*e^x + 25*x*e^(2*x) +
3*x)/(x^4 - 2*x^3*e^x - 10*x^3 + x^2*e^(2*x) + 20*x^2*e^x + 25*x^2 - 10*x*e^(2*x) - 50*x*e^x + 25*e^(2*x) + 5)

Mupad [F(-1)]

Timed out. \[ \int \frac {-20-275 x^2+120 x^3-638 x^4+500 x^5-150 x^6+20 x^7-x^8+e^{4 x} \left (-625+500 x-150 x^2+20 x^3-x^4\right )+e^{3 x} \left (2500 x-2000 x^2+600 x^3-80 x^4+4 x^5\right )+e^{2 x} \left (-225+50 x-3741 x^2+2998 x^3-900 x^4+120 x^5-6 x^6\right )+e^x \left (500 x-170 x^2+2504 x^3-1998 x^4+600 x^5-80 x^6+4 x^7\right )}{25+250 x^2-100 x^3+635 x^4-500 x^5+150 x^6-20 x^7+x^8+e^{4 x} \left (625-500 x+150 x^2-20 x^3+x^4\right )+e^{3 x} \left (-2500 x+2000 x^2-600 x^3+80 x^4-4 x^5\right )+e^{2 x} \left (250-100 x+3760 x^2-3000 x^3+900 x^4-120 x^5+6 x^6\right )+e^x \left (-500 x+200 x^2-2520 x^3+2000 x^4-600 x^5+80 x^6-4 x^7\right )} \, dx=\int -\frac {{\mathrm {e}}^{4\,x}\,\left (x^4-20\,x^3+150\,x^2-500\,x+625\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x^5-80\,x^4+600\,x^3-2000\,x^2+2500\,x\right )-{\mathrm {e}}^x\,\left (4\,x^7-80\,x^6+600\,x^5-1998\,x^4+2504\,x^3-170\,x^2+500\,x\right )+275\,x^2-120\,x^3+638\,x^4-500\,x^5+150\,x^6-20\,x^7+x^8+{\mathrm {e}}^{2\,x}\,\left (6\,x^6-120\,x^5+900\,x^4-2998\,x^3+3741\,x^2-50\,x+225\right )+20}{{\mathrm {e}}^{4\,x}\,\left (x^4-20\,x^3+150\,x^2-500\,x+625\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x^5-80\,x^4+600\,x^3-2000\,x^2+2500\,x\right )-{\mathrm {e}}^x\,\left (4\,x^7-80\,x^6+600\,x^5-2000\,x^4+2520\,x^3-200\,x^2+500\,x\right )+250\,x^2-100\,x^3+635\,x^4-500\,x^5+150\,x^6-20\,x^7+x^8+{\mathrm {e}}^{2\,x}\,\left (6\,x^6-120\,x^5+900\,x^4-3000\,x^3+3760\,x^2-100\,x+250\right )+25} \,d x \]

[In]

int(-(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5
) - exp(x)*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 275*x^2 - 120*x^3 + 638*x^4 -
500*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3741*x^2 - 50*x - 2998*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 225) + 2
0)/(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5)
- exp(x)*(500*x - 200*x^2 + 2520*x^3 - 2000*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 250*x^2 - 100*x^3 + 635*x^4 - 50
0*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3760*x^2 - 100*x - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 250) + 25
),x)

[Out]

int(-(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5
) - exp(x)*(500*x - 170*x^2 + 2504*x^3 - 1998*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 275*x^2 - 120*x^3 + 638*x^4 -
500*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3741*x^2 - 50*x - 2998*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 225) + 2
0)/(exp(4*x)*(150*x^2 - 500*x - 20*x^3 + x^4 + 625) - exp(3*x)*(2500*x - 2000*x^2 + 600*x^3 - 80*x^4 + 4*x^5)
- exp(x)*(500*x - 200*x^2 + 2520*x^3 - 2000*x^4 + 600*x^5 - 80*x^6 + 4*x^7) + 250*x^2 - 100*x^3 + 635*x^4 - 50
0*x^5 + 150*x^6 - 20*x^7 + x^8 + exp(2*x)*(3760*x^2 - 100*x - 3000*x^3 + 900*x^4 - 120*x^5 + 6*x^6 + 250) + 25
), x)