Integrand size = 99, antiderivative size = 26 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=x \left (-e^{2 x}+5 \left (x+2 e^3 x+x \log (4)\right )\right )^2 \] Output:
(10*x*ln(2)+10*x*exp(3)+5*x-exp(2*x))^2*x
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=x \left (e^{2 x}-10 e^3 x-5 x (1+\log (4))\right )^2 \] Input:
Integrate[75*x^2 + 300*E^3*x^2 + 300*E^6*x^2 + E^(4*x)*(1 + 4*x) + (150*x^ 2 + 300*E^3*x^2)*Log[4] + 75*x^2*Log[4]^2 + E^(2*x)*(-20*x - 20*x^2 + E^3* (-40*x - 40*x^2) + (-20*x - 20*x^2)*Log[4]),x]
Output:
x*(E^(2*x) - 10*E^3*x - 5*x*(1 + Log[4]))^2
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(26)=52\).
Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6, 6, 6, 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.
\(\displaystyle \int \left (300 e^6 x^2+300 e^3 x^2+75 x^2+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x^2+e^3 \left (-40 x^2-40 x\right )+\left (-20 x^2-20 x\right ) \log (4)-20 x\right )+\left (300 e^3 x^2+150 x^2\right ) \log (4)+e^{4 x} (4 x+1)\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (\left (75+300 e^3\right ) x^2+300 e^6 x^2+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x^2+e^3 \left (-40 x^2-40 x\right )+\left (-20 x^2-20 x\right ) \log (4)-20 x\right )+\left (300 e^3 x^2+150 x^2\right ) \log (4)+e^{4 x} (4 x+1)\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (\left (75+300 e^3+300 e^6\right ) x^2+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x^2+e^3 \left (-40 x^2-40 x\right )+\left (-20 x^2-20 x\right ) \log (4)-20 x\right )+\left (300 e^3 x^2+150 x^2\right ) \log (4)+e^{4 x} (4 x+1)\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (x^2 \left (75+300 e^3+300 e^6+75 \log ^2(4)\right )+e^{2 x} \left (-20 x^2+e^3 \left (-40 x^2-40 x\right )+\left (-20 x^2-20 x\right ) \log (4)-20 x\right )+\left (300 e^3 x^2+150 x^2\right ) \log (4)+e^{4 x} (4 x+1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 25 x^3 \left (1+4 e^3+4 e^6+\log ^2(4)\right )+50 \left (1+2 e^3\right ) x^3 \log (4)-10 e^{2 x} x^2-10 e^{2 x+3} x^2 \left (2+\frac {\log (4)}{e^3}\right )-\frac {e^{4 x}}{4}+\frac {1}{4} e^{4 x} (4 x+1)\) |
Input:
Int[75*x^2 + 300*E^3*x^2 + 300*E^6*x^2 + E^(4*x)*(1 + 4*x) + (150*x^2 + 30 0*E^3*x^2)*Log[4] + 75*x^2*Log[4]^2 + E^(2*x)*(-20*x - 20*x^2 + E^3*(-40*x - 40*x^2) + (-20*x - 20*x^2)*Log[4]),x]
Output:
-1/4*E^(4*x) - 10*E^(2*x)*x^2 + (E^(4*x)*(1 + 4*x))/4 + 50*(1 + 2*E^3)*x^3 *Log[4] - 10*E^(3 + 2*x)*x^2*(2 + Log[4]/E^3) + 25*x^3*(1 + 4*E^3 + 4*E^6 + Log[4]^2)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31
method | result | size |
norman | \(x \,{\mathrm e}^{4 x}+\left (100 \,{\mathrm e}^{6}+200 \,{\mathrm e}^{3} \ln \left (2\right )+100 \ln \left (2\right )^{2}+100 \,{\mathrm e}^{3}+100 \ln \left (2\right )+25\right ) x^{3}+\left (-20 \,{\mathrm e}^{3}-20 \ln \left (2\right )-10\right ) x^{2} {\mathrm e}^{2 x}\) | \(60\) |
risch | \(x \,{\mathrm e}^{4 x}-10 \left (2 \,{\mathrm e}^{3}+2 \ln \left (2\right )+1\right ) x^{2} {\mathrm e}^{2 x}+100 x^{3} \ln \left (2\right )^{2}+200 \ln \left (2\right ) x^{3} {\mathrm e}^{3}+100 x^{3} \ln \left (2\right )+100 \,{\mathrm e}^{6} x^{3}+100 x^{3} {\mathrm e}^{3}+25 x^{3}\) | \(71\) |
derivativedivides | \(100 \ln \left (2\right ) x^{3} \left (2 \,{\mathrm e}^{3}+1\right )+25 x^{3}+100 x^{3} {\mathrm e}^{3}+100 \,{\mathrm e}^{6} x^{3}+100 x^{3} \ln \left (2\right )^{2}+x \,{\mathrm e}^{4 x}-10 \,{\mathrm e}^{2 x} x^{2}-20 \,{\mathrm e}^{3} {\mathrm e}^{2 x} x^{2}-20 \ln \left (2\right ) {\mathrm e}^{2 x} x^{2}\) | \(84\) |
default | \(100 \ln \left (2\right ) x^{3} \left (2 \,{\mathrm e}^{3}+1\right )+25 x^{3}+100 x^{3} {\mathrm e}^{3}+100 \,{\mathrm e}^{6} x^{3}+100 x^{3} \ln \left (2\right )^{2}+x \,{\mathrm e}^{4 x}-10 \,{\mathrm e}^{2 x} x^{2}-20 \,{\mathrm e}^{3} {\mathrm e}^{2 x} x^{2}-20 \ln \left (2\right ) {\mathrm e}^{2 x} x^{2}\) | \(84\) |
parallelrisch | \(200 \ln \left (2\right ) x^{3} {\mathrm e}^{3}+100 x^{3} \ln \left (2\right )^{2}-20 \,{\mathrm e}^{3} {\mathrm e}^{2 x} x^{2}+100 x^{3} {\mathrm e}^{3}-20 \ln \left (2\right ) {\mathrm e}^{2 x} x^{2}+100 x^{3} \ln \left (2\right )+100 \,{\mathrm e}^{6} x^{3}-10 \,{\mathrm e}^{2 x} x^{2}+25 x^{3}+x \,{\mathrm e}^{4 x}\) | \(87\) |
parts | \(200 \ln \left (2\right ) x^{3} {\mathrm e}^{3}+100 x^{3} \ln \left (2\right )^{2}-20 \,{\mathrm e}^{3} {\mathrm e}^{2 x} x^{2}+100 x^{3} {\mathrm e}^{3}-20 \ln \left (2\right ) {\mathrm e}^{2 x} x^{2}+100 x^{3} \ln \left (2\right )+100 \,{\mathrm e}^{6} x^{3}-10 \,{\mathrm e}^{2 x} x^{2}+25 x^{3}+x \,{\mathrm e}^{4 x}\) | \(87\) |
orering | \(\frac {\left (32 x^{5}+64 x^{4}-52 x^{3}-14 x^{2}+26 x +1\right ) x \left (\left (1+4 x \right ) {\mathrm e}^{4 x}+\left (2 \left (-20 x^{2}-20 x \right ) \ln \left (2\right )+\left (-40 x^{2}-40 x \right ) {\mathrm e}^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{2 x}+300 x^{2} \ln \left (2\right )^{2}+2 \left (300 x^{2} {\mathrm e}^{3}+150 x^{2}\right ) \ln \left (2\right )+300 x^{2} {\mathrm e}^{6}+300 x^{2} {\mathrm e}^{3}+75 x^{2}\right )}{3 \left (-1+2 x \right ) \left (16 x^{4}+16 x^{3}-2 x^{2}-4 x +1\right )}-\frac {\left (64 x^{6}-32 x^{5}-32 x^{4}+32 x^{3}+6 x^{2}+2 x -1\right ) \left (4 \,{\mathrm e}^{4 x}+4 \left (1+4 x \right ) {\mathrm e}^{4 x}+\left (2 \left (-40 x -20\right ) \ln \left (2\right )+\left (-80 x -40\right ) {\mathrm e}^{3}-40 x -20\right ) {\mathrm e}^{2 x}+2 \left (2 \left (-20 x^{2}-20 x \right ) \ln \left (2\right )+\left (-40 x^{2}-40 x \right ) {\mathrm e}^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{2 x}+600 x \ln \left (2\right )^{2}+2 \left (600 x \,{\mathrm e}^{3}+300 x \right ) \ln \left (2\right )+600 x \,{\mathrm e}^{6}+600 x \,{\mathrm e}^{3}+150 x \right )}{8 \left (-1+2 x \right ) \left (16 x^{4}+16 x^{3}-2 x^{2}-4 x +1\right )}+\frac {\left (64 x^{6}-96 x^{5}+48 x^{4}+6 x^{2}-1\right ) \left (32 \,{\mathrm e}^{4 x}+16 \left (1+4 x \right ) {\mathrm e}^{4 x}+\left (-80 \ln \left (2\right )-80 \,{\mathrm e}^{3}-40\right ) {\mathrm e}^{2 x}+4 \left (2 \left (-40 x -20\right ) \ln \left (2\right )+\left (-80 x -40\right ) {\mathrm e}^{3}-40 x -20\right ) {\mathrm e}^{2 x}+4 \left (2 \left (-20 x^{2}-20 x \right ) \ln \left (2\right )+\left (-40 x^{2}-40 x \right ) {\mathrm e}^{3}-20 x^{2}-20 x \right ) {\mathrm e}^{2 x}+600 \ln \left (2\right )^{2}+2 \left (600 \,{\mathrm e}^{3}+300\right ) \ln \left (2\right )+600 \,{\mathrm e}^{6}+600 \,{\mathrm e}^{3}+150\right )}{48 \left (-1+2 x \right ) \left (16 x^{4}+16 x^{3}-2 x^{2}-4 x +1\right )}\) | \(528\) |
Input:
int((1+4*x)*exp(2*x)^2+(2*(-20*x^2-20*x)*ln(2)+(-40*x^2-40*x)*exp(3)-20*x^ 2-20*x)*exp(2*x)+300*x^2*ln(2)^2+2*(300*x^2*exp(3)+150*x^2)*ln(2)+300*x^2* exp(3)^2+300*x^2*exp(3)+75*x^2,x,method=_RETURNVERBOSE)
Output:
x*exp(2*x)^2+(100*exp(3)^2+200*exp(3)*ln(2)+100*ln(2)^2+100*exp(3)+100*ln( 2)+25)*x^3+(-20*exp(3)-20*ln(2)-10)*x^2*exp(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=100 \, x^{3} \log \left (2\right )^{2} + 100 \, x^{3} e^{6} + 100 \, x^{3} e^{3} + 25 \, x^{3} + x e^{\left (4 \, x\right )} - 10 \, {\left (2 \, x^{2} e^{3} + 2 \, x^{2} \log \left (2\right ) + x^{2}\right )} e^{\left (2 \, x\right )} + 100 \, {\left (2 \, x^{3} e^{3} + x^{3}\right )} \log \left (2\right ) \] Input:
integrate((1+4*x)*exp(2*x)^2+(2*(-20*x^2-20*x)*log(2)+(-40*x^2-40*x)*exp(3 )-20*x^2-20*x)*exp(2*x)+300*x^2*log(2)^2+2*(300*x^2*exp(3)+150*x^2)*log(2) +300*x^2*exp(3)^2+300*x^2*exp(3)+75*x^2,x, algorithm="fricas")
Output:
100*x^3*log(2)^2 + 100*x^3*e^6 + 100*x^3*e^3 + 25*x^3 + x*e^(4*x) - 10*(2* x^2*e^3 + 2*x^2*log(2) + x^2)*e^(2*x) + 100*(2*x^3*e^3 + x^3)*log(2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=x^{3} \cdot \left (25 + 100 \log {\left (2 \right )}^{2} + 100 \log {\left (2 \right )} + 100 e^{3} + 200 e^{3} \log {\left (2 \right )} + 100 e^{6}\right ) + x e^{4 x} + \left (- 20 x^{2} e^{3} - 20 x^{2} \log {\left (2 \right )} - 10 x^{2}\right ) e^{2 x} \] Input:
integrate((1+4*x)*exp(2*x)**2+(2*(-20*x**2-20*x)*ln(2)+(-40*x**2-40*x)*exp (3)-20*x**2-20*x)*exp(2*x)+300*x**2*ln(2)**2+2*(300*x**2*exp(3)+150*x**2)* ln(2)+300*x**2*exp(3)**2+300*x**2*exp(3)+75*x**2,x)
Output:
x**3*(25 + 100*log(2)**2 + 100*log(2) + 100*exp(3) + 200*exp(3)*log(2) + 1 00*exp(6)) + x*exp(4*x) + (-20*x**2*exp(3) - 20*x**2*log(2) - 10*x**2)*exp (2*x)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=100 \, x^{3} \log \left (2\right )^{2} + 100 \, x^{3} e^{6} + 100 \, x^{3} e^{3} - 10 \, x^{2} {\left (2 \, e^{3} + 2 \, \log \left (2\right ) + 1\right )} e^{\left (2 \, x\right )} + 25 \, x^{3} + x e^{\left (4 \, x\right )} + 100 \, {\left (2 \, x^{3} e^{3} + x^{3}\right )} \log \left (2\right ) \] Input:
integrate((1+4*x)*exp(2*x)^2+(2*(-20*x^2-20*x)*log(2)+(-40*x^2-40*x)*exp(3 )-20*x^2-20*x)*exp(2*x)+300*x^2*log(2)^2+2*(300*x^2*exp(3)+150*x^2)*log(2) +300*x^2*exp(3)^2+300*x^2*exp(3)+75*x^2,x, algorithm="maxima")
Output:
100*x^3*log(2)^2 + 100*x^3*e^6 + 100*x^3*e^3 - 10*x^2*(2*e^3 + 2*log(2) + 1)*e^(2*x) + 25*x^3 + x*e^(4*x) + 100*(2*x^3*e^3 + x^3)*log(2)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.00 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=100 \, x^{3} \log \left (2\right )^{2} + 100 \, x^{3} e^{6} + 100 \, x^{3} e^{3} + 25 \, x^{3} - 20 \, x^{2} e^{\left (2 \, x + 3\right )} + x e^{\left (4 \, x\right )} - 10 \, {\left (2 \, x^{2} \log \left (2\right ) + x^{2}\right )} e^{\left (2 \, x\right )} + 100 \, {\left (2 \, x^{3} e^{3} + x^{3}\right )} \log \left (2\right ) \] Input:
integrate((1+4*x)*exp(2*x)^2+(2*(-20*x^2-20*x)*log(2)+(-40*x^2-40*x)*exp(3 )-20*x^2-20*x)*exp(2*x)+300*x^2*log(2)^2+2*(300*x^2*exp(3)+150*x^2)*log(2) +300*x^2*exp(3)^2+300*x^2*exp(3)+75*x^2,x, algorithm="giac")
Output:
100*x^3*log(2)^2 + 100*x^3*e^6 + 100*x^3*e^3 + 25*x^3 - 20*x^2*e^(2*x + 3) + x*e^(4*x) - 10*(2*x^2*log(2) + x^2)*e^(2*x) + 100*(2*x^3*e^3 + x^3)*log (2)
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=x\,{\left (5\,x-{\mathrm {e}}^{2\,x}+10\,x\,{\mathrm {e}}^3+10\,x\,\ln \left (2\right )\right )}^2 \] Input:
int(300*x^2*log(2)^2 - exp(2*x)*(20*x + exp(3)*(40*x + 40*x^2) + 2*log(2)* (20*x + 20*x^2) + 20*x^2) + 300*x^2*exp(3) + 300*x^2*exp(6) + exp(4*x)*(4* x + 1) + 75*x^2 + 2*log(2)*(300*x^2*exp(3) + 150*x^2),x)
Output:
x*(5*x - exp(2*x) + 10*x*exp(3) + 10*x*log(2))^2
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.23 \[ \int \left (75 x^2+300 e^3 x^2+300 e^6 x^2+e^{4 x} (1+4 x)+\left (150 x^2+300 e^3 x^2\right ) \log (4)+75 x^2 \log ^2(4)+e^{2 x} \left (-20 x-20 x^2+e^3 \left (-40 x-40 x^2\right )+\left (-20 x-20 x^2\right ) \log (4)\right )\right ) \, dx=x \left (e^{4 x}-20 e^{2 x} \mathrm {log}\left (2\right ) x -20 e^{2 x} e^{3} x -10 e^{2 x} x +100 \mathrm {log}\left (2\right )^{2} x^{2}+200 \,\mathrm {log}\left (2\right ) e^{3} x^{2}+100 \,\mathrm {log}\left (2\right ) x^{2}+100 e^{6} x^{2}+100 e^{3} x^{2}+25 x^{2}\right ) \] Input:
int((1+4*x)*exp(2*x)^2+(2*(-20*x^2-20*x)*log(2)+(-40*x^2-40*x)*exp(3)-20*x ^2-20*x)*exp(2*x)+300*x^2*log(2)^2+2*(300*x^2*exp(3)+150*x^2)*log(2)+300*x ^2*exp(3)^2+300*x^2*exp(3)+75*x^2,x)
Output:
x*(e**(4*x) - 20*e**(2*x)*log(2)*x - 20*e**(2*x)*e**3*x - 10*e**(2*x)*x + 100*log(2)**2*x**2 + 200*log(2)*e**3*x**2 + 100*log(2)*x**2 + 100*e**6*x** 2 + 100*e**3*x**2 + 25*x**2)