Integrand size = 140, antiderivative size = 29 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\left (-e^{x^2}+\frac {5 e^{5+x-\frac {x^2}{25}}}{x}\right )^4 x \]
Time = 7.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\frac {e^{-\frac {4 x^2}{25}} \left (-5 e^{5+x}+e^{\frac {26 x^2}{25}} x\right )^4}{x^3} \]
Integrate[(E^(20 + 4*x)*(-9375 + 12500*x - 1000*x^2) + E^(15 + 3*x + (26*x ^2)/25)*(5000*x - 7500*x^2 - 4400*x^3) + E^(10 + 2*x + (52*x^2)/25)*(-750* x^2 + 1500*x^3 + 2880*x^4) + E^(5 + x + (78*x^2)/25)*(-100*x^4 - 592*x^5) + E^((104*x^2)/25)*(5*x^4 + 40*x^6))/(5*E^((4*x^2)/25)*x^4),x]
Leaf count is larger than twice the leaf count of optimal. \(134\) vs. \(2(29)=58\).
Time = 1.71 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.62, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {27, 25, 7239, 7293, 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.
\(\displaystyle \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{4 x+20} \left (-1000 x^2+12500 x-9375\right )+e^{\frac {26 x^2}{25}+3 x+15} \left (-4400 x^3-7500 x^2+5000 x\right )+e^{\frac {104 x^2}{25}} \left (40 x^6+5 x^4\right )+e^{\frac {78 x^2}{25}+x+5} \left (-592 x^5-100 x^4\right )+e^{\frac {52 x^2}{25}+2 x+10} \left (2880 x^4+1500 x^3-750 x^2\right )\right )}{5 x^4} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int -\frac {e^{-\frac {4 x^2}{25}} \left (125 e^{4 x+20} \left (8 x^2-100 x+75\right )-100 e^{\frac {26 x^2}{25}+3 x+15} \left (-44 x^3-75 x^2+50 x\right )+30 e^{\frac {52 x^2}{25}+2 x+10} \left (-96 x^4-50 x^3+25 x^2\right )+4 e^{\frac {78 x^2}{25}+x+5} \left (148 x^5+25 x^4\right )-5 e^{\frac {104 x^2}{25}} \left (8 x^6+x^4\right )\right )}{x^4}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {4 x^2}{25}} \left (125 e^{4 x+20} \left (8 x^2-100 x+75\right )-100 e^{\frac {26 x^2}{25}+3 x+15} \left (-44 x^3-75 x^2+50 x\right )+30 e^{\frac {52 x^2}{25}+2 x+10} \left (-96 x^4-50 x^3+25 x^2\right )+4 e^{\frac {78 x^2}{25}+x+5} \left (148 x^5+25 x^4\right )-5 e^{\frac {104 x^2}{25}} \left (8 x^6+x^4\right )\right )}{x^4}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {4 x^2}{25}} \left (5 e^{x+5}-e^{\frac {26 x^2}{25}} x\right )^3 \left (5 e^{\frac {26 x^2}{25}} x \left (8 x^2+1\right )+e^{x+5} \left (8 x^2-100 x+75\right )\right )}{x^4}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{5} \int \left (4 e^{\frac {74 x^2}{25}+x+5} (148 x+25)-5 e^{4 x^2} \left (8 x^2+1\right )+\frac {125 e^{-\frac {4 x^2}{25}+4 x+20} \left (8 x^2-100 x+75\right )}{x^4}+\frac {100 e^{\frac {22 x^2}{25}+3 x+15} \left (44 x^2+75 x-50\right )}{x^3}-\frac {30 e^{\frac {48 x^2}{25}+2 x+10} \left (96 x^2+50 x-25\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (5 e^{4 x^2} x-100 e^{\frac {74 x^2}{25}+x+5}+\frac {750 e^{\frac {48 x^2}{25}+2 x+10} \left (48 x^2+25 x\right )}{(48 x+25) x^2}+\frac {3125 e^{-\frac {4 x^2}{25}+4 x+20} \left (25 x-2 x^2\right )}{(25-2 x) x^4}-\frac {2500 e^{\frac {22 x^2}{25}+3 x+15} \left (44 x^2+75 x\right )}{(44 x+75) x^3}\right )\) |
Int[(E^(20 + 4*x)*(-9375 + 12500*x - 1000*x^2) + E^(15 + 3*x + (26*x^2)/25 )*(5000*x - 7500*x^2 - 4400*x^3) + E^(10 + 2*x + (52*x^2)/25)*(-750*x^2 + 1500*x^3 + 2880*x^4) + E^(5 + x + (78*x^2)/25)*(-100*x^4 - 592*x^5) + E^(( 104*x^2)/25)*(5*x^4 + 40*x^6))/(5*E^((4*x^2)/25)*x^4),x]
(-100*E^(5 + x + (74*x^2)/25) + 5*E^(4*x^2)*x + (3125*E^(20 + 4*x - (4*x^2 )/25)*(25*x - 2*x^2))/((25 - 2*x)*x^4) - (2500*E^(15 + 3*x + (22*x^2)/25)* (75*x + 44*x^2))/(x^3*(75 + 44*x)) + (750*E^(10 + 2*x + (48*x^2)/25)*(25*x + 48*x^2))/(x^2*(25 + 48*x)))/5
3.3.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(28)=56\).
Time = 2.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38
method | result | size |
risch | \(x \,{\mathrm e}^{4 x^{2}}-20 \,{\mathrm e}^{5+x +\frac {74}{25} x^{2}}+\frac {150 \,{\mathrm e}^{2 x +10+\frac {48}{25} x^{2}}}{x}-\frac {500 \,{\mathrm e}^{15+3 x +\frac {22}{25} x^{2}}}{x^{2}}+\frac {625 \,{\mathrm e}^{20+4 x -\frac {4}{25} x^{2}}}{x^{3}}\) | \(69\) |
parallelrisch | \(\frac {\left (125 \,{\mathrm e}^{4 x^{2}} {\mathrm e}^{\frac {4 x^{2}}{25}} x^{4}-2500 \,{\mathrm e}^{\frac {3 x^{2}}{25}} {\mathrm e}^{5+x} {\mathrm e}^{3 x^{2}} x^{3}+18750 \,{\mathrm e}^{2 x^{2}} {\mathrm e}^{\frac {2 x^{2}}{25}} {\mathrm e}^{2 x +10} x^{2}-62500 \,{\mathrm e}^{15+3 x} x \,{\mathrm e}^{\frac {x^{2}}{25}} {\mathrm e}^{x^{2}}+78125 \,{\mathrm e}^{20+4 x}\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{125 x^{3}}\) | \(109\) |
int(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4)*exp( 5+x)*exp(1/25*x^2)^3*exp(x^2)^3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x)^2*exp (1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp(1/25*x^ 2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4,x,met hod=_RETURNVERBOSE)
x*exp(4*x^2)-20*exp(5+x+74/25*x^2)+150/x*exp(2*x+10+48/25*x^2)-500/x^2*exp (15+3*x+22/25*x^2)+625/x^3*exp(20+4*x-4/25*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\frac {{\left (x^{4} - 20 \, x^{3} e^{\left (-\frac {26}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (-\frac {52}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (-\frac {78}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {104}{25} \, x^{2} + 4 \, x + 20\right )}\right )} e^{\left (4 \, x^{2}\right )}}{x^{3}} \]
integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4 )*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x) ^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp(1 /25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4 ,x, algorithm=\
(x^4 - 20*x^3*e^(-26/25*x^2 + x + 5) + 150*x^2*e^(-52/25*x^2 + 2*x + 10) - 500*x*e^(-78/25*x^2 + 3*x + 15) + 625*e^(-104/25*x^2 + 4*x + 20))*e^(4*x^ 2)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).
Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=x e^{4 x^{2}} + \frac {\left (- 20 x^{6} e^{\frac {78 x^{2}}{25}} e^{x + 5} + 150 x^{5} e^{\frac {52 x^{2}}{25}} e^{2 x + 10} - 500 x^{4} e^{\frac {26 x^{2}}{25}} e^{3 x + 15} + 625 x^{3} e^{4 x + 20}\right ) e^{- \frac {4 x^{2}}{25}}}{x^{6}} \]
integrate(1/5*((40*x**6+5*x**4)*exp(1/25*x**2)**4*exp(x**2)**4+(-592*x**5- 100*x**4)*exp(5+x)*exp(1/25*x**2)**3*exp(x**2)**3+(2880*x**4+1500*x**3-750 *x**2)*exp(5+x)**2*exp(1/25*x**2)**2*exp(x**2)**2+(-4400*x**3-7500*x**2+50 00*x)*exp(5+x)**3*exp(1/25*x**2)*exp(x**2)+(-1000*x**2+12500*x-9375)*exp(5 +x)**4)/x**4/exp(1/25*x**2)**4,x)
x*exp(4*x**2) + (-20*x**6*exp(78*x**2/25)*exp(x + 5) + 150*x**5*exp(52*x** 2/25)*exp(2*x + 10) - 500*x**4*exp(26*x**2/25)*exp(3*x + 15) + 625*x**3*ex p(4*x + 20))*exp(-4*x**2/25)/x**6
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\frac {x^{4} e^{\left (4 \, x^{2}\right )} - 20 \, x^{3} e^{\left (\frac {74}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (\frac {48}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (\frac {22}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {4}{25} \, x^{2} + 4 \, x + 20\right )}}{x^{3}} \]
integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4 )*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x) ^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp(1 /25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4 ,x, algorithm=\
(x^4*e^(4*x^2) - 20*x^3*e^(74/25*x^2 + x + 5) + 150*x^2*e^(48/25*x^2 + 2*x + 10) - 500*x*e^(22/25*x^2 + 3*x + 15) + 625*e^(-4/25*x^2 + 4*x + 20))/x^ 3
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\frac {x^{4} e^{\left (4 \, x^{2}\right )} - 20 \, x^{3} e^{\left (\frac {74}{25} \, x^{2} + x + 5\right )} + 150 \, x^{2} e^{\left (\frac {48}{25} \, x^{2} + 2 \, x + 10\right )} - 500 \, x e^{\left (\frac {22}{25} \, x^{2} + 3 \, x + 15\right )} + 625 \, e^{\left (-\frac {4}{25} \, x^{2} + 4 \, x + 20\right )}}{x^{3}} \]
integrate(1/5*((40*x^6+5*x^4)*exp(1/25*x^2)^4*exp(x^2)^4+(-592*x^5-100*x^4 )*exp(5+x)*exp(1/25*x^2)^3*exp(x^2)^3+(2880*x^4+1500*x^3-750*x^2)*exp(5+x) ^2*exp(1/25*x^2)^2*exp(x^2)^2+(-4400*x^3-7500*x^2+5000*x)*exp(5+x)^3*exp(1 /25*x^2)*exp(x^2)+(-1000*x^2+12500*x-9375)*exp(5+x)^4)/x^4/exp(1/25*x^2)^4 ,x, algorithm=\
(x^4*e^(4*x^2) - 20*x^3*e^(74/25*x^2 + x + 5) + 150*x^2*e^(48/25*x^2 + 2*x + 10) - 500*x*e^(22/25*x^2 + 3*x + 15) + 625*e^(-4/25*x^2 + 4*x + 20))/x^ 3
Time = 13.51 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34 \[ \int \frac {e^{-\frac {4 x^2}{25}} \left (e^{20+4 x} \left (-9375+12500 x-1000 x^2\right )+e^{15+3 x+\frac {26 x^2}{25}} \left (5000 x-7500 x^2-4400 x^3\right )+e^{10+2 x+\frac {52 x^2}{25}} \left (-750 x^2+1500 x^3+2880 x^4\right )+e^{5+x+\frac {78 x^2}{25}} \left (-100 x^4-592 x^5\right )+e^{\frac {104 x^2}{25}} \left (5 x^4+40 x^6\right )\right )}{5 x^4} \, dx=\frac {625\,{\mathrm {e}}^{-\frac {4\,x^2}{25}+4\,x+20}}{x^3}-20\,{\mathrm {e}}^{\frac {74\,x^2}{25}+x+5}-\frac {500\,{\mathrm {e}}^{\frac {22\,x^2}{25}+3\,x+15}}{x^2}+\frac {150\,{\mathrm {e}}^{\frac {48\,x^2}{25}+2\,x+10}}{x}+x\,{\mathrm {e}}^{4\,x^2} \]
int(-(exp(-(4*x^2)/25)*((exp(4*x + 20)*(1000*x^2 - 12500*x + 9375))/5 - (e xp(4*x^2)*exp((4*x^2)/25)*(5*x^4 + 40*x^6))/5 - (exp(2*x + 10)*exp(2*x^2)* exp((2*x^2)/25)*(1500*x^3 - 750*x^2 + 2880*x^4))/5 + (exp(x^2)*exp(3*x + 1 5)*exp(x^2/25)*(7500*x^2 - 5000*x + 4400*x^3))/5 + (exp(x + 5)*exp(3*x^2)* exp((3*x^2)/25)*(100*x^4 + 592*x^5))/5))/x^4,x)