Integrand size = 138, antiderivative size = 33 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {1}{25} e^2 (-4-x)^2 \left (-\frac {e}{5}+e^{2 x}-x\right )^2 x^2 \] Output:
1/25*(-4-x)^2*(exp(x)^2-x-1/5*exp(1))^2*x^2*exp(2)
Time = 1.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {1}{625} e^2 x^2 (4+x)^2 \left (e-5 e^{2 x}+5 x\right )^2 \] Input:
Integrate[(E^(2 + 4*x)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4) + E^(2 + 2*x )*(-2400*x^2 - 3200*x^3 - 1050*x^4 - 100*x^5 + E*(-320*x - 560*x^2 - 200*x ^3 - 20*x^4)) + E^2*(1600*x^3 + 1000*x^4 + 150*x^5 + E^2*(32*x + 24*x^2 + 4*x^3) + E*(480*x^2 + 320*x^3 + 50*x^4)))/625,x]
Output:
(E^2*x^2*(4 + x)^2*(E - 5*E^(2*x) + 5*x)^2)/625
Leaf count is larger than twice the leaf count of optimal. \(184\) vs. \(2(33)=66\).
Time = 1.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 5.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {27, 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.
| \(\displaystyle \int \frac {1}{625} \left (e^{4 x+2} \left (100 x^4+900 x^3+2200 x^2+800 x\right )+e^{2 x+2} \left (-100 x^5-1050 x^4-3200 x^3-2400 x^2+e \left (-20 x^4-200 x^3-560 x^2-320 x\right )\right )+e^2 \left (150 x^5+1000 x^4+1600 x^3+e^2 \left (4 x^3+24 x^2+32 x\right )+e \left (50 x^4+320 x^3+480 x^2\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{625} \int \left (100 e^{4 x+2} \left (x^4+9 x^3+22 x^2+8 x\right )-10 e^{2 x+2} \left (10 x^5+105 x^4+320 x^3+240 x^2+2 e \left (x^4+10 x^3+28 x^2+16 x\right )\right )+2 e^2 \left (75 x^5+500 x^4+800 x^3+2 e^2 \left (x^3+6 x^2+8 x\right )+5 e \left (5 x^4+32 x^3+48 x^2\right )\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{625} \left (25 e^2 x^6-50 e^{2 x+2} x^5+10 e^3 x^5+200 e^2 x^5-400 e^{2 x+2} x^4-10 e^{2 x+3} x^4+25 e^{4 x+2} x^4+e^4 x^4+80 e^3 x^4+400 e^2 x^4-800 e^{2 x+2} x^3-80 e^{2 x+3} x^3+200 e^{4 x+2} x^3+8 e^4 x^3+160 e^3 x^3-160 e^{2 x+3} x^2+400 e^{4 x+2} x^2+16 e^4 x^2\right )\) |
Input:
Int[(E^(2 + 4*x)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4) + E^(2 + 2*x)*(-24 00*x^2 - 3200*x^3 - 1050*x^4 - 100*x^5 + E*(-320*x - 560*x^2 - 200*x^3 - 2 0*x^4)) + E^2*(1600*x^3 + 1000*x^4 + 150*x^5 + E^2*(32*x + 24*x^2 + 4*x^3) + E*(480*x^2 + 320*x^3 + 50*x^4)))/625,x]
Output:
(16*E^4*x^2 - 160*E^(3 + 2*x)*x^2 + 400*E^(2 + 4*x)*x^2 + 160*E^3*x^3 + 8* E^4*x^3 - 800*E^(2 + 2*x)*x^3 - 80*E^(3 + 2*x)*x^3 + 200*E^(2 + 4*x)*x^3 + 400*E^2*x^4 + 80*E^3*x^4 + E^4*x^4 - 400*E^(2 + 2*x)*x^4 - 10*E^(3 + 2*x) *x^4 + 25*E^(2 + 4*x)*x^4 + 200*E^2*x^5 + 10*E^3*x^5 - 50*E^(2 + 2*x)*x^5 + 25*E^2*x^6)/625
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(28)=56\).
Time = 0.73 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.36
| method | result | size |
| risch | \(\frac {\left (25 x^{4}+200 x^{3}+400 x^{2}\right ) {\mathrm e}^{4 x +2}}{625}+\frac {\left (-10 x^{4} {\mathrm e}-50 x^{5}-80 x^{3} {\mathrm e}-400 x^{4}-160 x^{2} {\mathrm e}-800 x^{3}\right ) {\mathrm e}^{2+2 x}}{625}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}}{125}+\frac {8 \,{\mathrm e}^{2} x^{5}}{25}+\frac {x^{4} {\mathrm e}^{4}}{625}+\frac {16 \,{\mathrm e}^{2} {\mathrm e} x^{4}}{125}+\frac {16 x^{4} {\mathrm e}^{2}}{25}+\frac {8 x^{3} {\mathrm e}^{4}}{625}+\frac {32 \,{\mathrm e}^{2} {\mathrm e} x^{3}}{125}+\frac {16 x^{2} {\mathrm e}^{4}}{625}\) | \(144\) |
| norman | \(\left (\frac {2 \,{\mathrm e}^{2} {\mathrm e}}{125}+\frac {8 \,{\mathrm e}^{2}}{25}\right ) x^{5}+\left (\frac {8 \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {32 \,{\mathrm e}^{2} {\mathrm e}}{125}\right ) x^{3}+\left (\frac {\left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 \,{\mathrm e}^{2} {\mathrm e}}{125}+\frac {16 \,{\mathrm e}^{2}}{25}\right ) x^{4}+\left (-\frac {32 \,{\mathrm e}^{2}}{25}-\frac {16 \,{\mathrm e}^{2} {\mathrm e}}{125}\right ) x^{3} {\mathrm e}^{2 x}+\left (-\frac {16 \,{\mathrm e}^{2}}{25}-\frac {2 \,{\mathrm e}^{2} {\mathrm e}}{125}\right ) x^{4} {\mathrm e}^{2 x}+\frac {x^{6} {\mathrm e}^{2}}{25}+\frac {16 x^{2} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {16 x^{2} \left ({\mathrm e}^{2}\right )^{2}}{625}-\frac {2 \,{\mathrm e}^{2} {\mathrm e}^{2 x} x^{5}}{25}+\frac {8 \,{\mathrm e}^{2} {\mathrm e}^{4 x} x^{3}}{25}+\frac {{\mathrm e}^{2} {\mathrm e}^{4 x} x^{4}}{25}-\frac {32 \,{\mathrm e}^{2} {\mathrm e}^{2 x} {\mathrm e} x^{2}}{125}\) | \(172\) |
| parallelrisch | \(-\frac {32 \,{\mathrm e}^{2} {\mathrm e}^{2 x} {\mathrm e} x^{2}}{125}-\frac {16 \,{\mathrm e}^{2} {\mathrm e}^{2 x} {\mathrm e} x^{3}}{125}-\frac {32 \,{\mathrm e}^{2 x} x^{3} {\mathrm e}^{2}}{25}-\frac {2 \,{\mathrm e}^{2} {\mathrm e}^{2 x} {\mathrm e} x^{4}}{125}-\frac {16 \,{\mathrm e}^{2} {\mathrm e}^{2 x} x^{4}}{25}-\frac {2 \,{\mathrm e}^{2} {\mathrm e}^{2 x} x^{5}}{25}+\frac {16 x^{2} {\mathrm e}^{2} {\mathrm e}^{4 x}}{25}+\frac {8 \,{\mathrm e}^{2} {\mathrm e}^{4 x} x^{3}}{25}+\frac {{\mathrm e}^{2} {\mathrm e}^{4 x} x^{4}}{25}+\frac {16 x^{2} \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {8 x^{3} \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {32 \,{\mathrm e}^{2} {\mathrm e} x^{3}}{125}+\frac {x^{4} \left ({\mathrm e}^{2}\right )^{2}}{625}+\frac {16 \,{\mathrm e}^{2} {\mathrm e} x^{4}}{125}+\frac {16 x^{4} {\mathrm e}^{2}}{25}+\frac {2 \,{\mathrm e}^{2} x^{5} {\mathrm e}}{125}+\frac {8 \,{\mathrm e}^{2} x^{5}}{25}+\frac {x^{6} {\mathrm e}^{2}}{25}\) | \(188\) |
| parts | \(\frac {2 \,{\mathrm e}^{2} \left (\frac {25 x^{6}}{2}+\frac {\left (500+25 \,{\mathrm e}\right ) x^{5}}{5}+\frac {x^{4} \left (2 \,{\mathrm e}^{2}+160 \,{\mathrm e}+800\right )}{4}+\frac {x^{3} \left (12 \,{\mathrm e}^{2}+240 \,{\mathrm e}\right )}{3}+8 x^{2} {\mathrm e}^{2}\right )}{625}-\frac {2 \,{\mathrm e}^{2} \left (80 \,{\mathrm e}^{2 x} x^{3}+40 \,{\mathrm e}^{2 x} x^{4}+5 x^{5} {\mathrm e}^{2 x}+56 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )+20 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )+32 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )+2 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\right )}{125}+\frac {4 \,{\mathrm e}^{2} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}+2 x^{3} {\mathrm e}^{4 x}+4 x^{2} {\mathrm e}^{4 x}\right )}{25}\) | \(255\) |
| default | \(\frac {{\mathrm e}^{2} \left (25 x^{6}+\frac {2 \left (500+25 \,{\mathrm e}\right ) x^{5}}{5}+\frac {\left (60 \,{\mathrm e}+800+\left (100+2 \,{\mathrm e}\right ) {\mathrm e}\right ) x^{4}}{2}+\frac {2 \left (80 \,{\mathrm e}+\left (12 \,{\mathrm e}+160\right ) {\mathrm e}\right ) x^{3}}{3}+16 x^{2} {\mathrm e}^{2}\right )}{625}+\frac {{\mathrm e}^{2} \left (-800 \,{\mathrm e}^{2 x} x^{3}-400 \,{\mathrm e}^{2 x} x^{4}-50 x^{5} {\mathrm e}^{2 x}-560 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )-200 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )-320 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-20 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{4}}{2}-{\mathrm e}^{2 x} x^{3}+\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {3 x \,{\mathrm e}^{2 x}}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4}\right )\right )}{625}+\frac {4 \,{\mathrm e}^{2} \left (\frac {x^{4} {\mathrm e}^{4 x}}{4}+2 x^{3} {\mathrm e}^{4 x}+4 x^{2} {\mathrm e}^{4 x}\right )}{25}\) | \(261\) |
Input:
int(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4 -200*x^3-560*x^2-320*x)*exp(1)-100*x^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)* exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2)*exp( 1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x,method=_RETURNVERBOSE)
Output:
1/625*(25*x^4+200*x^3+400*x^2)*exp(4*x+2)+1/625*(-10*x^4*exp(1)-50*x^5-80* x^3*exp(1)-400*x^4-160*x^2*exp(1)-800*x^3)*exp(2+2*x)+1/25*x^6*exp(2)+2/12 5*exp(2)*x^5*exp(1)+8/25*exp(2)*x^5+1/625*x^4*exp(2)^2+16/125*exp(2)*exp(1 )*x^4+16/25*x^4*exp(2)+8/625*x^3*exp(2)^2+32/125*exp(2)*exp(1)*x^3+16/625* x^2*exp(4)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (26) = 52\).
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.76 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {1}{625} \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{6} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{5} + 25 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} e^{4} + 25 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 4\right )} - 10 \, {\left ({\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{3} + 5 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{2}\right )} e^{\left (2 \, x + 2\right )}\right )} e^{\left (-2\right )} \] Input:
integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((- 20*x^4-200*x^3-560*x^2-320*x)*exp(1)-100*x^5-1050*x^4-3200*x^3-2400*x^2)*e xp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2 )*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="fricas")
Output:
1/625*((x^4 + 8*x^3 + 16*x^2)*e^6 + 10*(x^5 + 8*x^4 + 16*x^3)*e^5 + 25*(x^ 6 + 8*x^5 + 16*x^4)*e^4 + 25*(x^4 + 8*x^3 + 16*x^2)*e^(4*x + 4) - 10*((x^4 + 8*x^3 + 16*x^2)*e^3 + 5*(x^5 + 8*x^4 + 16*x^3)*e^2)*e^(2*x + 2))*e^(-2)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (27) = 54\).
Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.00 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {x^{6} e^{2}}{25} + x^{5} \cdot \left (\frac {2 e^{3}}{125} + \frac {8 e^{2}}{25}\right ) + x^{4} \left (\frac {e^{4}}{625} + \frac {16 e^{3}}{125} + \frac {16 e^{2}}{25}\right ) + x^{3} \cdot \left (\frac {8 e^{4}}{625} + \frac {32 e^{3}}{125}\right ) + \frac {16 x^{2} e^{4}}{625} + \frac {\left (125 x^{4} e^{2} + 1000 x^{3} e^{2} + 2000 x^{2} e^{2}\right ) e^{4 x}}{3125} + \frac {\left (- 250 x^{5} e^{2} - 2000 x^{4} e^{2} - 50 x^{4} e^{3} - 4000 x^{3} e^{2} - 400 x^{3} e^{3} - 800 x^{2} e^{3}\right ) e^{2 x}}{3125} \] Input:
integrate(1/625*(100*x**4+900*x**3+2200*x**2+800*x)*exp(2)*exp(x)**4+1/625 *((-20*x**4-200*x**3-560*x**2-320*x)*exp(1)-100*x**5-1050*x**4-3200*x**3-2 400*x**2)*exp(2)*exp(x)**2+1/625*((4*x**3+24*x**2+32*x)*exp(1)**2+(50*x**4 +320*x**3+480*x**2)*exp(1)+150*x**5+1000*x**4+1600*x**3)*exp(2),x)
Output:
x**6*exp(2)/25 + x**5*(2*exp(3)/125 + 8*exp(2)/25) + x**4*(exp(4)/625 + 16 *exp(3)/125 + 16*exp(2)/25) + x**3*(8*exp(4)/625 + 32*exp(3)/125) + 16*x** 2*exp(4)/625 + (125*x**4*exp(2) + 1000*x**3*exp(2) + 2000*x**2*exp(2))*exp (4*x)/3125 + (-250*x**5*exp(2) - 2000*x**4*exp(2) - 50*x**4*exp(3) - 4000* x**3*exp(2) - 400*x**3*exp(3) - 800*x**2*exp(3))*exp(2*x)/3125
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (26) = 52\).
Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.85 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} e^{2} + 8 \, x^{3} e^{2} + 16 \, x^{2} e^{2}\right )} e^{\left (4 \, x\right )} - \frac {2}{125} \, {\left (5 \, x^{5} e^{2} + x^{4} {\left (e^{3} + 40 \, e^{2}\right )} + 8 \, x^{3} {\left (e^{3} + 10 \, e^{2}\right )} + 16 \, x^{2} e^{3}\right )} e^{\left (2 \, x\right )} \] Input:
integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((- 20*x^4-200*x^3-560*x^2-320*x)*exp(1)-100*x^5-1050*x^4-3200*x^3-2400*x^2)*e xp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2 )*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="maxima")
Output:
1/625*(25*x^6 + 200*x^5 + 400*x^4 + (x^4 + 8*x^3 + 16*x^2)*e^2 + 10*(x^5 + 8*x^4 + 16*x^3)*e)*e^2 + 1/25*(x^4*e^2 + 8*x^3*e^2 + 16*x^2*e^2)*e^(4*x) - 2/125*(5*x^5*e^2 + x^4*(e^3 + 40*e^2) + 8*x^3*(e^3 + 10*e^2) + 16*x^2*e^ 3)*e^(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.70 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {1}{625} \, {\left (25 \, x^{6} + 200 \, x^{5} + 400 \, x^{4} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{2} + 10 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e\right )} e^{2} + \frac {1}{25} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (4 \, x + 2\right )} - \frac {2}{125} \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (2 \, x + 3\right )} - \frac {2}{25} \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} e^{\left (2 \, x + 2\right )} \] Input:
integrate(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((- 20*x^4-200*x^3-560*x^2-320*x)*exp(1)-100*x^5-1050*x^4-3200*x^3-2400*x^2)*e xp(2)*exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2 )*exp(1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x, algorithm="giac")
Output:
1/625*(25*x^6 + 200*x^5 + 400*x^4 + (x^4 + 8*x^3 + 16*x^2)*e^2 + 10*(x^5 + 8*x^4 + 16*x^3)*e)*e^2 + 1/25*(x^4 + 8*x^3 + 16*x^2)*e^(4*x + 2) - 2/125* (x^4 + 8*x^3 + 16*x^2)*e^(2*x + 3) - 2/25*(x^5 + 8*x^4 + 16*x^3)*e^(2*x + 2)
Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.45 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=x^5\,\left (\frac {8\,{\mathrm {e}}^2}{25}+\frac {2\,{\mathrm {e}}^3}{125}\right )+x^3\,\left (\frac {32\,{\mathrm {e}}^3}{125}+\frac {8\,{\mathrm {e}}^4}{625}\right )+\frac {16\,x^2\,{\mathrm {e}}^4}{625}+\frac {x^6\,{\mathrm {e}}^2}{25}-\frac {32\,x^2\,{\mathrm {e}}^{2\,x+3}}{125}+\frac {16\,x^2\,{\mathrm {e}}^{4\,x+2}}{25}+\frac {8\,x^3\,{\mathrm {e}}^{4\,x+2}}{25}-\frac {2\,x^5\,{\mathrm {e}}^{2\,x+2}}{25}+\frac {x^4\,{\mathrm {e}}^{4\,x+2}}{25}+x^4\,\left (\frac {16\,{\mathrm {e}}^2}{25}+\frac {16\,{\mathrm {e}}^3}{125}+\frac {{\mathrm {e}}^4}{625}\right )-\frac {x^4\,{\mathrm {e}}^{2\,x+2}\,\left (10\,\mathrm {e}+400\right )}{625}-\frac {x^3\,{\mathrm {e}}^{2\,x+2}\,\left (80\,\mathrm {e}+800\right )}{625} \] Input:
int((exp(2)*(exp(2)*(32*x + 24*x^2 + 4*x^3) + exp(1)*(480*x^2 + 320*x^3 + 50*x^4) + 1600*x^3 + 1000*x^4 + 150*x^5))/625 - (exp(2*x)*exp(2)*(exp(1)*( 320*x + 560*x^2 + 200*x^3 + 20*x^4) + 2400*x^2 + 3200*x^3 + 1050*x^4 + 100 *x^5))/625 + (exp(4*x)*exp(2)*(800*x + 2200*x^2 + 900*x^3 + 100*x^4))/625, x)
Output:
x^5*((8*exp(2))/25 + (2*exp(3))/125) + x^3*((32*exp(3))/125 + (8*exp(4))/6 25) + (16*x^2*exp(4))/625 + (x^6*exp(2))/25 - (32*x^2*exp(2*x + 3))/125 + (16*x^2*exp(4*x + 2))/25 + (8*x^3*exp(4*x + 2))/25 - (2*x^5*exp(2*x + 2))/ 25 + (x^4*exp(4*x + 2))/25 + x^4*((16*exp(2))/25 + (16*exp(3))/125 + exp(4 )/625) - (x^4*exp(2*x + 2)*(10*exp(1) + 400))/625 - (x^3*exp(2*x + 2)*(80* exp(1) + 800))/625
Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.21 \[ \int \frac {1}{625} \left (e^{2+4 x} \left (800 x+2200 x^2+900 x^3+100 x^4\right )+e^{2+2 x} \left (-2400 x^2-3200 x^3-1050 x^4-100 x^5+e \left (-320 x-560 x^2-200 x^3-20 x^4\right )\right )+e^2 \left (1600 x^3+1000 x^4+150 x^5+e^2 \left (32 x+24 x^2+4 x^3\right )+e \left (480 x^2+320 x^3+50 x^4\right )\right )\right ) \, dx=\frac {e^{2} x^{2} \left (25 e^{4 x} x^{2}+200 e^{4 x} x +400 e^{4 x}-10 e^{2 x} e \,x^{2}-80 e^{2 x} e x -160 e^{2 x} e -50 e^{2 x} x^{3}-400 e^{2 x} x^{2}-800 e^{2 x} x +e^{2} x^{2}+8 e^{2} x +16 e^{2}+10 e \,x^{3}+80 e \,x^{2}+160 e x +25 x^{4}+200 x^{3}+400 x^{2}\right )}{625} \] Input:
int(1/625*(100*x^4+900*x^3+2200*x^2+800*x)*exp(2)*exp(x)^4+1/625*((-20*x^4 -200*x^3-560*x^2-320*x)*exp(1)-100*x^5-1050*x^4-3200*x^3-2400*x^2)*exp(2)* exp(x)^2+1/625*((4*x^3+24*x^2+32*x)*exp(1)^2+(50*x^4+320*x^3+480*x^2)*exp( 1)+150*x^5+1000*x^4+1600*x^3)*exp(2),x)
Output:
(e**2*x**2*(25*e**(4*x)*x**2 + 200*e**(4*x)*x + 400*e**(4*x) - 10*e**(2*x) *e*x**2 - 80*e**(2*x)*e*x - 160*e**(2*x)*e - 50*e**(2*x)*x**3 - 400*e**(2* x)*x**2 - 800*e**(2*x)*x + e**2*x**2 + 8*e**2*x + 16*e**2 + 10*e*x**3 + 80 *e*x**2 + 160*e*x + 25*x**4 + 200*x**3 + 400*x**2))/625