Integrand size = 92, antiderivative size = 30 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^2 (4+x)^2 \left (4+e^{x/2}+625 (-5+x)-x^2\right )^2 \] Output:
x^2*(4+x)^2*(exp(1/2*x)-x^2+625*x-3121)^2
Time = 2.42 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^2 (4+x)^2 \left (3121-e^{x/2}-625 x+x^2\right )^2 \] Input:
Integrate[311700512*x + 46515384*x^2 - 60477948*x^3 - 3731570*x^4 + 232129 8*x^5 - 8694*x^6 + 8*x^7 + E^x*(32*x + 40*x^2 + 12*x^3 + x^4) + E^(x/2)*(- 199744*x - 139744*x^2 - 64*x^3 + 8033*x^4 + 605*x^5 - x^6),x]
Output:
x^2*(4 + x)^2*(3121 - E^(x/2) - 625*x + x^2)^2
Leaf count is larger than twice the leaf count of optimal. \(117\) vs. \(2(30)=60\).
Time = 0.57 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.90, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {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.
\(\displaystyle \int \left (8 x^7-8694 x^6+2321298 x^5-3731570 x^4-60477948 x^3+46515384 x^2+e^x \left (x^4+12 x^3+40 x^2+32 x\right )+e^{x/2} \left (-x^6+605 x^5+8033 x^4-64 x^3-139744 x^2-199744 x\right )+311700512 x\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^8-1242 x^7-2 e^{x/2} x^6+386883 x^6+1234 e^{x/2} x^5-746314 x^5+3726 e^{x/2} x^4+e^x x^4-15119487 x^4-29936 e^{x/2} x^3+8 e^x x^3+15505128 x^3-99872 e^{x/2} x^2+16 e^x x^2+155850256 x^2\) |
Input:
Int[311700512*x + 46515384*x^2 - 60477948*x^3 - 3731570*x^4 + 2321298*x^5 - 8694*x^6 + 8*x^7 + E^x*(32*x + 40*x^2 + 12*x^3 + x^4) + E^(x/2)*(-199744 *x - 139744*x^2 - 64*x^3 + 8033*x^4 + 605*x^5 - x^6),x]
Output:
155850256*x^2 - 99872*E^(x/2)*x^2 + 16*E^x*x^2 + 15505128*x^3 - 29936*E^(x /2)*x^3 + 8*E^x*x^3 - 15119487*x^4 + 3726*E^(x/2)*x^4 + E^x*x^4 - 746314*x ^5 + 1234*E^(x/2)*x^5 + 386883*x^6 - 2*E^(x/2)*x^6 - 1242*x^7 + x^8
Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(25)=50\).
Time = 1.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77
method | result | size |
risch | \(\left (x^{4}+8 x^{3}+16 x^{2}\right ) {\mathrm e}^{x}+\left (-2 x^{6}+1234 x^{5}+3726 x^{4}-29936 x^{3}-99872 x^{2}\right ) {\mathrm e}^{\frac {x}{2}}+x^{8}-1242 x^{7}+386883 x^{6}-746314 x^{5}-15119487 x^{4}+15505128 x^{3}+155850256 x^{2}\) | \(83\) |
derivativedivides | \(x^{8}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}-1242 x^{7}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}+386883 x^{6}+{\mathrm e}^{x} x^{4}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}-746314 x^{5}+8 \,{\mathrm e}^{x} x^{3}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}-15119487 x^{4}+16 \,{\mathrm e}^{x} x^{2}-99872 \,{\mathrm e}^{\frac {x}{2}} x^{2}+15505128 x^{3}+155850256 x^{2}\) | \(112\) |
default | \(x^{8}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}-1242 x^{7}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}+386883 x^{6}+{\mathrm e}^{x} x^{4}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}-746314 x^{5}+8 \,{\mathrm e}^{x} x^{3}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}-15119487 x^{4}+16 \,{\mathrm e}^{x} x^{2}-99872 \,{\mathrm e}^{\frac {x}{2}} x^{2}+15505128 x^{3}+155850256 x^{2}\) | \(112\) |
norman | \(x^{8}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}-1242 x^{7}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}+386883 x^{6}+{\mathrm e}^{x} x^{4}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}-746314 x^{5}+8 \,{\mathrm e}^{x} x^{3}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}-15119487 x^{4}+16 \,{\mathrm e}^{x} x^{2}-99872 \,{\mathrm e}^{\frac {x}{2}} x^{2}+15505128 x^{3}+155850256 x^{2}\) | \(112\) |
parallelrisch | \(x^{8}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}-1242 x^{7}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}+386883 x^{6}+{\mathrm e}^{x} x^{4}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}-746314 x^{5}+8 \,{\mathrm e}^{x} x^{3}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}-15119487 x^{4}+16 \,{\mathrm e}^{x} x^{2}-99872 \,{\mathrm e}^{\frac {x}{2}} x^{2}+15505128 x^{3}+155850256 x^{2}\) | \(112\) |
parts | \(x^{8}-2 \,{\mathrm e}^{\frac {x}{2}} x^{6}-1242 x^{7}+1234 \,{\mathrm e}^{\frac {x}{2}} x^{5}+386883 x^{6}+{\mathrm e}^{x} x^{4}+3726 \,{\mathrm e}^{\frac {x}{2}} x^{4}-746314 x^{5}+8 \,{\mathrm e}^{x} x^{3}-29936 \,{\mathrm e}^{\frac {x}{2}} x^{3}-15119487 x^{4}+16 \,{\mathrm e}^{x} x^{2}-99872 \,{\mathrm e}^{\frac {x}{2}} x^{2}+15505128 x^{3}+155850256 x^{2}\) | \(112\) |
orering | \(\frac {\left (x^{18}-1823 x^{17}+1085688 x^{16}-197392935 x^{15}-8180359746 x^{14}-42126325725 x^{13}+777158665281 x^{12}+6549806871048 x^{11}-13315525873368 x^{10}-179213566814528 x^{9}+2419638356469856 x^{8}+41123802225128064 x^{7}+198974728407793536 x^{6}-227397301012684800 x^{5}-5726701061298008064 x^{4}-21107822794832240640 x^{3}-33189610352869785600 x^{2}-22505447041075445760 x -7006253980737208320\right ) \left (\left (x^{4}+12 x^{3}+40 x^{2}+32 x \right ) {\mathrm e}^{x}+\left (-x^{6}+605 x^{5}+8033 x^{4}-64 x^{3}-139744 x^{2}-199744 x \right ) {\mathrm e}^{\frac {x}{2}}+8 x^{7}-8694 x^{6}+2321298 x^{5}-3731570 x^{4}-60477948 x^{3}+46515384 x^{2}+311700512 x \right )}{2 \left (4 x^{11}-6803 x^{10}+3806250 x^{9}-687921412 x^{8}-6465184945 x^{7}+89109279336 x^{6}+556614893332 x^{5}-4021568718470 x^{4}-9626591735976 x^{3}+26112316374432 x^{2}+123188282123904 x +128272522461696\right ) x^{3} \left (4+x \right )^{3}}-\frac {3 \left (x^{16}-1855 x^{15}+1141294 x^{14}-229250383 x^{13}-2290599014 x^{12}+29025260801 x^{11}+265875495949 x^{10}-872678548132 x^{9}-11163576983504 x^{8}+70668597122592 x^{7}+2144012586081696 x^{6}+14224855579736064 x^{5}-16935325145128704 x^{4}-475081655099919360 x^{3}-1510692347193968640 x^{2}-1729490295490928640 x -583854498394767360\right ) \left (\left (4 x^{3}+36 x^{2}+80 x +32\right ) {\mathrm e}^{x}+\left (x^{4}+12 x^{3}+40 x^{2}+32 x \right ) {\mathrm e}^{x}+\left (-6 x^{5}+3025 x^{4}+32132 x^{3}-192 x^{2}-279488 x -199744\right ) {\mathrm e}^{\frac {x}{2}}+\frac {\left (-x^{6}+605 x^{5}+8033 x^{4}-64 x^{3}-139744 x^{2}-199744 x \right ) {\mathrm e}^{\frac {x}{2}}}{2}+56 x^{6}-52164 x^{5}+11606490 x^{4}-14926280 x^{3}-181433844 x^{2}+93030768 x +311700512\right )}{2 \left (4 x^{11}-6803 x^{10}+3806250 x^{9}-687921412 x^{8}-6465184945 x^{7}+89109279336 x^{6}+556614893332 x^{5}-4021568718470 x^{4}-9626591735976 x^{3}+26112316374432 x^{2}+123188282123904 x +128272522461696\right ) x^{2} \left (4+x \right )^{2}}+\frac {\left (x^{14}-1871 x^{13}+1169940 x^{12}-246334663 x^{11}+1114408508 x^{10}+22706082345 x^{9}-57395952677 x^{8}-849972198128 x^{7}+91067929877664 x^{5}+1233333287305056 x^{4}+1013810207172480 x^{3}-45455012870837760 x^{2}-124959021562183680 x -109472718449018880\right ) \left (\left (12 x^{2}+72 x +80\right ) {\mathrm e}^{x}+2 \left (4 x^{3}+36 x^{2}+80 x +32\right ) {\mathrm e}^{x}+\left (x^{4}+12 x^{3}+40 x^{2}+32 x \right ) {\mathrm e}^{x}+\left (-30 x^{4}+12100 x^{3}+96396 x^{2}-384 x -279488\right ) {\mathrm e}^{\frac {x}{2}}+\left (-6 x^{5}+3025 x^{4}+32132 x^{3}-192 x^{2}-279488 x -199744\right ) {\mathrm e}^{\frac {x}{2}}+\frac {\left (-x^{6}+605 x^{5}+8033 x^{4}-64 x^{3}-139744 x^{2}-199744 x \right ) {\mathrm e}^{\frac {x}{2}}}{4}+336 x^{5}-260820 x^{4}+46425960 x^{3}-44778840 x^{2}-362867688 x +93030768\right )}{\left (4+x \right ) x \left (x^{2}-629 x +4371\right ) \left (4 x^{9}-4287 x^{8}+1092243 x^{7}+17837912 x^{6}-19332450 x^{5}-1020345066 x^{4}-680014232 x^{3}+10630613088 x^{2}+32406104448 x +29346264576\right )}\) | \(843\) |
Input:
int((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-13 9744*x^2-199744*x)*exp(1/2*x)+8*x^7-8694*x^6+2321298*x^5-3731570*x^4-60477 948*x^3+46515384*x^2+311700512*x,x,method=_RETURNVERBOSE)
Output:
(x^4+8*x^3+16*x^2)*exp(x)+(-2*x^6+1234*x^5+3726*x^4-29936*x^3-99872*x^2)*e xp(1/2*x)+x^8-1242*x^7+386883*x^6-746314*x^5-15119487*x^4+15505128*x^3+155 850256*x^2
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \] Input:
integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64* x^3-139744*x^2-199744*x)*exp(1/2*x)+8*x^7-8694*x^6+2321298*x^5-3731570*x^4 -60477948*x^3+46515384*x^2+311700512*x,x, algorithm="fricas")
Output:
x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 1 55850256*x^2 - 2*(x^6 - 617*x^5 - 1863*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2 *x) + (x^4 + 8*x^3 + 16*x^2)*e^x
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^{8} - 1242 x^{7} + 386883 x^{6} - 746314 x^{5} - 15119487 x^{4} + 15505128 x^{3} + 155850256 x^{2} + \left (x^{4} + 8 x^{3} + 16 x^{2}\right ) e^{x} + \left (- 2 x^{6} + 1234 x^{5} + 3726 x^{4} - 29936 x^{3} - 99872 x^{2}\right ) e^{\frac {x}{2}} \] Input:
integrate((x**4+12*x**3+40*x**2+32*x)*exp(1/2*x)**2+(-x**6+605*x**5+8033*x **4-64*x**3-139744*x**2-199744*x)*exp(1/2*x)+8*x**7-8694*x**6+2321298*x**5 -3731570*x**4-60477948*x**3+46515384*x**2+311700512*x,x)
Output:
x**8 - 1242*x**7 + 386883*x**6 - 746314*x**5 - 15119487*x**4 + 15505128*x* *3 + 155850256*x**2 + (x**4 + 8*x**3 + 16*x**2)*exp(x) + (-2*x**6 + 1234*x **5 + 3726*x**4 - 29936*x**3 - 99872*x**2)*exp(x/2)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \] Input:
integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64* x^3-139744*x^2-199744*x)*exp(1/2*x)+8*x^7-8694*x^6+2321298*x^5-3731570*x^4 -60477948*x^3+46515384*x^2+311700512*x,x, algorithm="maxima")
Output:
x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 1 55850256*x^2 - 2*(x^6 - 617*x^5 - 1863*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2 *x) + (x^4 + 8*x^3 + 16*x^2)*e^x
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^{8} - 1242 \, x^{7} + 386883 \, x^{6} - 746314 \, x^{5} - 15119487 \, x^{4} + 15505128 \, x^{3} + 155850256 \, x^{2} - 2 \, {\left (x^{6} - 617 \, x^{5} - 1863 \, x^{4} + 14968 \, x^{3} + 49936 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x\right )} + {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )} e^{x} \] Input:
integrate((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64* x^3-139744*x^2-199744*x)*exp(1/2*x)+8*x^7-8694*x^6+2321298*x^5-3731570*x^4 -60477948*x^3+46515384*x^2+311700512*x,x, algorithm="giac")
Output:
x^8 - 1242*x^7 + 386883*x^6 - 746314*x^5 - 15119487*x^4 + 15505128*x^3 + 1 55850256*x^2 - 2*(x^6 - 617*x^5 - 1863*x^4 + 14968*x^3 + 49936*x^2)*e^(1/2 *x) + (x^4 + 8*x^3 + 16*x^2)*e^x
Time = 2.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^2\,{\left (x+4\right )}^2\,{\left (625\,x+{\mathrm {e}}^{x/2}-x^2-3121\right )}^2 \] Input:
int(311700512*x - exp(x/2)*(199744*x + 139744*x^2 + 64*x^3 - 8033*x^4 - 60 5*x^5 + x^6) + exp(x)*(32*x + 40*x^2 + 12*x^3 + x^4) + 46515384*x^2 - 6047 7948*x^3 - 3731570*x^4 + 2321298*x^5 - 8694*x^6 + 8*x^7,x)
Output:
x^2*(x + 4)^2*(625*x + exp(x/2) - x^2 - 3121)^2
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \left (311700512 x+46515384 x^2-60477948 x^3-3731570 x^4+2321298 x^5-8694 x^6+8 x^7+e^x \left (32 x+40 x^2+12 x^3+x^4\right )+e^{x/2} \left (-199744 x-139744 x^2-64 x^3+8033 x^4+605 x^5-x^6\right )\right ) \, dx=x^{2} \left (-2 e^{\frac {x}{2}} x^{4}+1234 e^{\frac {x}{2}} x^{3}+3726 e^{\frac {x}{2}} x^{2}-29936 e^{\frac {x}{2}} x -99872 e^{\frac {x}{2}}+e^{x} x^{2}+8 e^{x} x +16 e^{x}+x^{6}-1242 x^{5}+386883 x^{4}-746314 x^{3}-15119487 x^{2}+15505128 x +155850256\right ) \] Input:
int((x^4+12*x^3+40*x^2+32*x)*exp(1/2*x)^2+(-x^6+605*x^5+8033*x^4-64*x^3-13 9744*x^2-199744*x)*exp(1/2*x)+8*x^7-8694*x^6+2321298*x^5-3731570*x^4-60477 948*x^3+46515384*x^2+311700512*x,x)
Output:
x**2*( - 2*e**(x/2)*x**4 + 1234*e**(x/2)*x**3 + 3726*e**(x/2)*x**2 - 29936 *e**(x/2)*x - 99872*e**(x/2) + e**x*x**2 + 8*e**x*x + 16*e**x + x**6 - 124 2*x**5 + 386883*x**4 - 746314*x**3 - 15119487*x**2 + 15505128*x + 15585025 6)