Integrand size = 130, antiderivative size = 28 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=5 \left (-5 e^2+e^{4 x} x+\frac {25}{(2+x)^2 (15+\log (4))^2}\right ) \] Output:
5*x*exp(4*x)+125/(2*ln(2)+15)^2/(2+x)^2-25*exp(2)
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=5 e^{4 x} x+\frac {125}{(2+x)^2 (15+\log (4))^2} \] Input:
Integrate[(-250 + E^(4*x)*(9000 + 49500*x + 60750*x^2 + 28125*x^3 + 4500*x ^4 + (1200 + 6600*x + 8100*x^2 + 3750*x^3 + 600*x^4)*Log[4] + (40 + 220*x + 270*x^2 + 125*x^3 + 20*x^4)*Log[4]^2))/(1800 + 2700*x + 1350*x^2 + 225*x ^3 + (240 + 360*x + 180*x^2 + 30*x^3)*Log[4] + (8 + 12*x + 6*x^2 + x^3)*Lo g[4]^2),x]
Output:
5*E^(4*x)*x + 125/((2 + x)^2*(15 + Log[4])^2)
Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2007, 7239, 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.
| \(\displaystyle \int \frac {e^{4 x} \left (4500 x^4+28125 x^3+60750 x^2+\left (20 x^4+125 x^3+270 x^2+220 x+40\right ) \log ^2(4)+\left (600 x^4+3750 x^3+8100 x^2+6600 x+1200\right ) \log (4)+49500 x+9000\right )-250}{225 x^3+1350 x^2+\left (x^3+6 x^2+12 x+8\right ) \log ^2(4)+\left (30 x^3+180 x^2+360 x+240\right ) \log (4)+2700 x+1800} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^{4 x} \left (4500 x^4+28125 x^3+60750 x^2+\left (20 x^4+125 x^3+270 x^2+220 x+40\right ) \log ^2(4)+\left (600 x^4+3750 x^3+8100 x^2+6600 x+1200\right ) \log (4)+49500 x+9000\right )-250}{\left (x (15+\log (4))^{2/3}+2 (15+\log (4))^{2/3}\right )^3}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (5 e^{4 x} (4 x+1)-\frac {250}{(x+2)^3 (15+\log (4))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{4} e^{4 x} (4 x+1)-\frac {5 e^{4 x}}{4}+\frac {125}{(x+2)^2 (15+\log (4))^2}\) |
Input:
Int[(-250 + E^(4*x)*(9000 + 49500*x + 60750*x^2 + 28125*x^3 + 4500*x^4 + ( 1200 + 6600*x + 8100*x^2 + 3750*x^3 + 600*x^4)*Log[4] + (40 + 220*x + 270* x^2 + 125*x^3 + 20*x^4)*Log[4]^2))/(1800 + 2700*x + 1350*x^2 + 225*x^3 + ( 240 + 360*x + 180*x^2 + 30*x^3)*Log[4] + (8 + 12*x + 6*x^2 + x^3)*Log[4]^2 ),x]
Output:
(-5*E^(4*x))/4 + (5*E^(4*x)*(1 + 4*x))/4 + 125/((2 + x)^2*(15 + Log[4])^2)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
| method | result | size |
| parts | \(5 x \,{\mathrm e}^{4 x}+\frac {125}{\left (4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225\right ) \left (2+x \right )^{2}}\) | \(30\) |
| risch | \(\frac {125}{4 \left (x^{2} \ln \left (2\right )^{2}+4 x \ln \left (2\right )^{2}+15 x^{2} \ln \left (2\right )+4 \ln \left (2\right )^{2}+60 x \ln \left (2\right )+\frac {225 x^{2}}{4}+60 \ln \left (2\right )+225 x +225\right )}+5 x \,{\mathrm e}^{4 x}\) | \(60\) |
| norman | \(\frac {\left (10 \ln \left (2\right )+75\right ) x^{3} {\mathrm e}^{4 x}+\left (40 \ln \left (2\right )+300\right ) x \,{\mathrm e}^{4 x}+\left (40 \ln \left (2\right )+300\right ) x^{2} {\mathrm e}^{4 x}+\frac {125}{2 \ln \left (2\right )+15}}{\left (2+x \right )^{2} \left (2 \ln \left (2\right )+15\right )}\) | \(66\) |
| parallelrisch | \(\frac {20 \ln \left (2\right )^{2} {\mathrm e}^{4 x} x^{3}+80 \ln \left (2\right )^{2} {\mathrm e}^{4 x} x^{2}+300 \ln \left (2\right ) {\mathrm e}^{4 x} x^{3}+80 \ln \left (2\right )^{2} {\mathrm e}^{4 x} x +1200 \ln \left (2\right ) {\mathrm e}^{4 x} x^{2}+1125 \,{\mathrm e}^{4 x} x^{3}+1200 \ln \left (2\right ) {\mathrm e}^{4 x} x +4500 \,{\mathrm e}^{4 x} x^{2}+125+4500 x \,{\mathrm e}^{4 x}}{\left (4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225\right ) \left (x^{2}+4 x +4\right )}\) | \(121\) |
| derivativedivides | \(\frac {2000}{\left (4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225\right ) \left (4 x +8\right )^{2}}+\frac {-\frac {72000 \,{\mathrm e}^{4 x}}{\left (4 x +8\right )^{2}}-\frac {72000 \,{\mathrm e}^{4 x}}{4 x +8}-72000 \,{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {72000 \,{\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {72000 \,{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {72000 \,{\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {1125 \,{\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}+\frac {38400 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )^{2}}-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )}-\frac {{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{2}\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {2560 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )^{2}}-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )}-\frac {{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{2}\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {300 \ln \left (2\right ) {\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {20 \ln \left (2\right )^{2} {\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}\) | \(445\) |
| default | \(\frac {2000}{\left (4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225\right ) \left (4 x +8\right )^{2}}+\frac {-\frac {72000 \,{\mathrm e}^{4 x}}{\left (4 x +8\right )^{2}}-\frac {72000 \,{\mathrm e}^{4 x}}{4 x +8}-72000 \,{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {72000 \,{\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {72000 \,{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {72000 \,{\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {1125 \,{\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}+\frac {38400 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )^{2}}-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )}-\frac {{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{2}\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {2560 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )^{2}}-\frac {{\mathrm e}^{4 x}}{2 \left (4 x +8\right )}-\frac {{\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{2}\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {19200 \ln \left (2\right ) {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {300 \ln \left (2\right ) {\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{-8} \operatorname {expIntegral}_{1}\left (-4 x -8\right )}{\left (2 \ln \left (2\right )+15\right )^{2}}+\frac {1280 \ln \left (2\right )^{2} {\mathrm e}^{4 x}}{\left (2 \ln \left (2\right )+15\right )^{2} \left (4 x +8\right )^{2}}+\frac {20 \ln \left (2\right )^{2} {\mathrm e}^{4 x} x}{4 \ln \left (2\right )^{2}+60 \ln \left (2\right )+225}\) | \(445\) |
| orering | \(-\frac {\left (8 x^{3}+36 x^{2}+45 x +16\right ) \left (\left (4 \left (20 x^{4}+125 x^{3}+270 x^{2}+220 x +40\right ) \ln \left (2\right )^{2}+2 \left (600 x^{4}+3750 x^{3}+8100 x^{2}+6600 x +1200\right ) \ln \left (2\right )+4500 x^{4}+28125 x^{3}+60750 x^{2}+49500 x +9000\right ) {\mathrm e}^{4 x}-250\right )}{\left (16 x^{2}+52 x +19\right ) \left (4 \left (x^{3}+6 x^{2}+12 x +8\right ) \ln \left (2\right )^{2}+2 \left (30 x^{3}+180 x^{2}+360 x +240\right ) \ln \left (2\right )+225 x^{3}+1350 x^{2}+2700 x +1800\right )}+\frac {\left (2+x \right ) \left (4 x^{2}+11 x +2\right ) \left (\frac {\left (4 \left (80 x^{3}+375 x^{2}+540 x +220\right ) \ln \left (2\right )^{2}+2 \left (2400 x^{3}+11250 x^{2}+16200 x +6600\right ) \ln \left (2\right )+18000 x^{3}+84375 x^{2}+121500 x +49500\right ) {\mathrm e}^{4 x}+4 \left (4 \left (20 x^{4}+125 x^{3}+270 x^{2}+220 x +40\right ) \ln \left (2\right )^{2}+2 \left (600 x^{4}+3750 x^{3}+8100 x^{2}+6600 x +1200\right ) \ln \left (2\right )+4500 x^{4}+28125 x^{3}+60750 x^{2}+49500 x +9000\right ) {\mathrm e}^{4 x}}{4 \left (x^{3}+6 x^{2}+12 x +8\right ) \ln \left (2\right )^{2}+2 \left (30 x^{3}+180 x^{2}+360 x +240\right ) \ln \left (2\right )+225 x^{3}+1350 x^{2}+2700 x +1800}-\frac {\left (\left (4 \left (20 x^{4}+125 x^{3}+270 x^{2}+220 x +40\right ) \ln \left (2\right )^{2}+2 \left (600 x^{4}+3750 x^{3}+8100 x^{2}+6600 x +1200\right ) \ln \left (2\right )+4500 x^{4}+28125 x^{3}+60750 x^{2}+49500 x +9000\right ) {\mathrm e}^{4 x}-250\right ) \left (4 \left (3 x^{2}+12 x +12\right ) \ln \left (2\right )^{2}+2 \left (90 x^{2}+360 x +360\right ) \ln \left (2\right )+675 x^{2}+2700 x +2700\right )}{{\left (4 \left (x^{3}+6 x^{2}+12 x +8\right ) \ln \left (2\right )^{2}+2 \left (30 x^{3}+180 x^{2}+360 x +240\right ) \ln \left (2\right )+225 x^{3}+1350 x^{2}+2700 x +1800\right )}^{2}}\right )}{32 x^{2}+104 x +38}\) | \(558\) |
Input:
int(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*ln(2)^2+2*(600*x^4+3750*x^3+8100 *x^2+6600*x+1200)*ln(2)+4500*x^4+28125*x^3+60750*x^2+49500*x+9000)*exp(4*x )-250)/(4*(x^3+6*x^2+12*x+8)*ln(2)^2+2*(30*x^3+180*x^2+360*x+240)*ln(2)+22 5*x^3+1350*x^2+2700*x+1800),x,method=_RETURNVERBOSE)
Output:
5*x*exp(4*x)+125/(4*ln(2)^2+60*ln(2)+225)/(2+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.39 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=\frac {5 \, {\left ({\left (225 \, x^{3} + 4 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (2\right )^{2} + 900 \, x^{2} + 60 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 900 \, x\right )} e^{\left (4 \, x\right )} + 25\right )}}{4 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (2\right )^{2} + 225 \, x^{2} + 60 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (2\right ) + 900 \, x + 900} \] Input:
integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x ^3+8100*x^2+6600*x+1200)*log(2)+4500*x^4+28125*x^3+60750*x^2+49500*x+9000) *exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+240) *log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="fricas")
Output:
5*((225*x^3 + 4*(x^3 + 4*x^2 + 4*x)*log(2)^2 + 900*x^2 + 60*(x^3 + 4*x^2 + 4*x)*log(2) + 900*x)*e^(4*x) + 25)/(4*(x^2 + 4*x + 4)*log(2)^2 + 225*x^2 + 60*(x^2 + 4*x + 4)*log(2) + 900*x + 900)
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=5 x e^{4 x} + \frac {250}{x^{2} \cdot \left (8 \log {\left (2 \right )}^{2} + 120 \log {\left (2 \right )} + 450\right ) + x \left (32 \log {\left (2 \right )}^{2} + 480 \log {\left (2 \right )} + 1800\right ) + 32 \log {\left (2 \right )}^{2} + 480 \log {\left (2 \right )} + 1800} \] Input:
integrate(((4*(20*x**4+125*x**3+270*x**2+220*x+40)*ln(2)**2+2*(600*x**4+37 50*x**3+8100*x**2+6600*x+1200)*ln(2)+4500*x**4+28125*x**3+60750*x**2+49500 *x+9000)*exp(4*x)-250)/(4*(x**3+6*x**2+12*x+8)*ln(2)**2+2*(30*x**3+180*x** 2+360*x+240)*ln(2)+225*x**3+1350*x**2+2700*x+1800),x)
Output:
5*x*exp(4*x) + 250/(x**2*(8*log(2)**2 + 120*log(2) + 450) + x*(32*log(2)** 2 + 480*log(2) + 1800) + 32*log(2)**2 + 480*log(2) + 1800)
\[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=\int { \frac {5 \, {\left ({\left (900 \, x^{4} + 5625 \, x^{3} + 4 \, {\left (4 \, x^{4} + 25 \, x^{3} + 54 \, x^{2} + 44 \, x + 8\right )} \log \left (2\right )^{2} + 12150 \, x^{2} + 60 \, {\left (4 \, x^{4} + 25 \, x^{3} + 54 \, x^{2} + 44 \, x + 8\right )} \log \left (2\right ) + 9900 \, x + 1800\right )} e^{\left (4 \, x\right )} - 50\right )}}{225 \, x^{3} + 4 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} \log \left (2\right )^{2} + 1350 \, x^{2} + 60 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} \log \left (2\right ) + 2700 \, x + 1800} \,d x } \] Input:
integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x ^3+8100*x^2+6600*x+1200)*log(2)+4500*x^4+28125*x^3+60750*x^2+49500*x+9000) *exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+240) *log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="maxima")
Output:
5*x*e^(4*x) - 160*e^(-8)*exp_integral_e(3, -4*x - 8)*log(2)^2/((x + 2)^2*( 2*log(2) + 15)^2) + 125/((4*log(2)^2 + 60*log(2) + 225)*x^2 + 4*(4*log(2)^ 2 + 60*log(2) + 225)*x + 16*log(2)^2 + 240*log(2) + 900) - 2400*e^(-8)*exp _integral_e(3, -4*x - 8)*log(2)/((x + 2)^2*(2*log(2) + 15)^2) - 9000*e^(-8 )*exp_integral_e(3, -4*x - 8)/((x + 2)^2*(2*log(2) + 15)^2) - 40*integrate (e^(4*x)/(x^3 + 6*x^2 + 12*x + 8), x)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 5.25 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=\frac {5 \, {\left (4 \, x^{3} e^{\left (4 \, x\right )} \log \left (2\right )^{2} + 60 \, x^{3} e^{\left (4 \, x\right )} \log \left (2\right ) + 16 \, x^{2} e^{\left (4 \, x\right )} \log \left (2\right )^{2} + 225 \, x^{3} e^{\left (4 \, x\right )} + 240 \, x^{2} e^{\left (4 \, x\right )} \log \left (2\right ) + 16 \, x e^{\left (4 \, x\right )} \log \left (2\right )^{2} + 900 \, x^{2} e^{\left (4 \, x\right )} + 240 \, x e^{\left (4 \, x\right )} \log \left (2\right ) + 900 \, x e^{\left (4 \, x\right )} + 25\right )}}{4 \, x^{2} \log \left (2\right )^{2} + 60 \, x^{2} \log \left (2\right ) + 16 \, x \log \left (2\right )^{2} + 225 \, x^{2} + 240 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2} + 900 \, x + 240 \, \log \left (2\right ) + 900} \] Input:
integrate(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x ^3+8100*x^2+6600*x+1200)*log(2)+4500*x^4+28125*x^3+60750*x^2+49500*x+9000) *exp(4*x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+240) *log(2)+225*x^3+1350*x^2+2700*x+1800),x, algorithm="giac")
Output:
5*(4*x^3*e^(4*x)*log(2)^2 + 60*x^3*e^(4*x)*log(2) + 16*x^2*e^(4*x)*log(2)^ 2 + 225*x^3*e^(4*x) + 240*x^2*e^(4*x)*log(2) + 16*x*e^(4*x)*log(2)^2 + 900 *x^2*e^(4*x) + 240*x*e^(4*x)*log(2) + 900*x*e^(4*x) + 25)/(4*x^2*log(2)^2 + 60*x^2*log(2) + 16*x*log(2)^2 + 225*x^2 + 240*x*log(2) + 16*log(2)^2 + 9 00*x + 240*log(2) + 900)
Time = 3.74 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=\frac {125}{\left (30\,\ln \left (4\right )+{\ln \left (4\right )}^2+225\right )\,x^2+\left (120\,\ln \left (4\right )+4\,{\ln \left (4\right )}^2+900\right )\,x+120\,\ln \left (4\right )+4\,{\ln \left (4\right )}^2+900}+5\,x\,{\mathrm {e}}^{4\,x} \] Input:
int((exp(4*x)*(49500*x + 4*log(2)^2*(220*x + 270*x^2 + 125*x^3 + 20*x^4 + 40) + 2*log(2)*(6600*x + 8100*x^2 + 3750*x^3 + 600*x^4 + 1200) + 60750*x^2 + 28125*x^3 + 4500*x^4 + 9000) - 250)/(2700*x + 2*log(2)*(360*x + 180*x^2 + 30*x^3 + 240) + 4*log(2)^2*(12*x + 6*x^2 + x^3 + 8) + 1350*x^2 + 225*x^ 3 + 1800),x)
Output:
125/(120*log(4) + x*(120*log(4) + 4*log(4)^2 + 900) + x^2*(30*log(4) + log (4)^2 + 225) + 4*log(4)^2 + 900) + 5*x*exp(4*x)
Time = 0.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 5.57 \[ \int \frac {-250+e^{4 x} \left (9000+49500 x+60750 x^2+28125 x^3+4500 x^4+\left (1200+6600 x+8100 x^2+3750 x^3+600 x^4\right ) \log (4)+\left (40+220 x+270 x^2+125 x^3+20 x^4\right ) \log ^2(4)\right )}{1800+2700 x+1350 x^2+225 x^3+\left (240+360 x+180 x^2+30 x^3\right ) \log (4)+\left (8+12 x+6 x^2+x^3\right ) \log ^2(4)} \, dx=\frac {20 e^{4 x} \mathrm {log}\left (2\right )^{2} x^{3}+80 e^{4 x} \mathrm {log}\left (2\right )^{2} x^{2}+80 e^{4 x} \mathrm {log}\left (2\right )^{2} x +300 e^{4 x} \mathrm {log}\left (2\right ) x^{3}+1200 e^{4 x} \mathrm {log}\left (2\right ) x^{2}+1200 e^{4 x} \mathrm {log}\left (2\right ) x +1125 e^{4 x} x^{3}+4500 e^{4 x} x^{2}+4500 e^{4 x} x +125}{4 \mathrm {log}\left (2\right )^{2} x^{2}+16 \mathrm {log}\left (2\right )^{2} x +16 \mathrm {log}\left (2\right )^{2}+60 \,\mathrm {log}\left (2\right ) x^{2}+240 \,\mathrm {log}\left (2\right ) x +240 \,\mathrm {log}\left (2\right )+225 x^{2}+900 x +900} \] Input:
int(((4*(20*x^4+125*x^3+270*x^2+220*x+40)*log(2)^2+2*(600*x^4+3750*x^3+810 0*x^2+6600*x+1200)*log(2)+4500*x^4+28125*x^3+60750*x^2+49500*x+9000)*exp(4 *x)-250)/(4*(x^3+6*x^2+12*x+8)*log(2)^2+2*(30*x^3+180*x^2+360*x+240)*log(2 )+225*x^3+1350*x^2+2700*x+1800),x)
Output:
(5*(4*e**(4*x)*log(2)**2*x**3 + 16*e**(4*x)*log(2)**2*x**2 + 16*e**(4*x)*l og(2)**2*x + 60*e**(4*x)*log(2)*x**3 + 240*e**(4*x)*log(2)*x**2 + 240*e**( 4*x)*log(2)*x + 225*e**(4*x)*x**3 + 900*e**(4*x)*x**2 + 900*e**(4*x)*x + 2 5))/(4*log(2)**2*x**2 + 16*log(2)**2*x + 16*log(2)**2 + 60*log(2)*x**2 + 2 40*log(2)*x + 240*log(2) + 225*x**2 + 900*x + 900)