Integrand size = 207, antiderivative size = 31 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=x+\frac {1}{2} \left (-x+\frac {x}{5 e+4 x^2 (5+x) \left (e^x+x\right )}\right ) \]
Time = 7.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\frac {1}{2} x \left (1+\frac {1}{5 e+4 e^x x^2 (5+x)+4 x^3 (5+x)}\right ) \]
Integrate[(25*E^2 - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16*x^8 + E*(5 + 200*x^3 + 40*x^4) + E^(2*x)*(400*x^4 + 160*x^5 + 16*x^6) + E^x*(-20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7 + E*(200*x^2 + 40*x^3)))/(50*E ^2 + 800*x^6 + 320*x^7 + 32*x^8 + E*(400*x^3 + 80*x^4) + E^(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + E^x*(1600*x^5 + 640*x^6 + 64*x^7 + E*(400*x^2 + 80*x ^3))),x]
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.
| \(\displaystyle \int \frac {16 x^8+160 x^7+400 x^6-12 x^4-40 x^3+e \left (40 x^4+200 x^3+5\right )+e^{2 x} \left (16 x^6+160 x^5+400 x^4\right )+e^x \left (32 x^7+320 x^6+800 x^5-4 x^4-28 x^3-20 x^2+e \left (40 x^3+200 x^2\right )\right )+25 e^2}{32 x^8+320 x^7+800 x^6+e \left (80 x^4+400 x^3\right )+e^{2 x} \left (32 x^6+320 x^5+800 x^4\right )+e^x \left (64 x^7+640 x^6+1600 x^5+e \left (80 x^3+400 x^2\right )\right )+50 e^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {16 x^8+160 x^7+400 x^6-12 x^4-40 x^3+e \left (40 x^4+200 x^3+5\right )+e^{2 x} \left (16 x^6+160 x^5+400 x^4\right )+e^x \left (32 x^7+320 x^6+800 x^5-4 x^4-28 x^3-20 x^2+e \left (40 x^3+200 x^2\right )\right )+25 e^2}{2 \left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {16 x^8+160 x^7+400 x^6-12 x^4-40 x^3+5 e \left (8 x^4+40 x^3+1\right )+16 e^{2 x} \left (x^6+10 x^5+25 x^4\right )-4 e^x \left (-8 x^7-80 x^6-200 x^5+x^4+7 x^3+5 x^2-10 e \left (x^3+5 x^2\right )\right )+25 e^2}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {x^2+7 x+5}{(x+5) \left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )}+\frac {4 x^6+36 x^5+60 x^4-100 x^3+5 e x^2+40 e x+50 e}{(x+5) \left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (15 e \int \frac {1}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx+5 e \int \frac {x}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx-20 \int \frac {x^3}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx+16 \int \frac {x^4}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx-25 e \int \frac {1}{(x+5) \left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx-2 \int \frac {1}{4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e}dx-\int \frac {x}{4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e}dx+5 \int \frac {1}{(x+5) \left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )}dx+4 \int \frac {x^5}{\left (4 x^4+4 e^x x^3+20 x^3+20 e^x x^2+5 e\right )^2}dx+x\right )\) |
Int[(25*E^2 - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16*x^8 + E*(5 + 200*x^ 3 + 40*x^4) + E^(2*x)*(400*x^4 + 160*x^5 + 16*x^6) + E^x*(-20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7 + E*(200*x^2 + 40*x^3)))/(50*E^2 + 8 00*x^6 + 320*x^7 + 32*x^8 + E*(400*x^3 + 80*x^4) + E^(2*x)*(800*x^4 + 320* x^5 + 32*x^6) + E^x*(1600*x^5 + 640*x^6 + 64*x^7 + E*(400*x^2 + 80*x^3))), x]
3.7.41.3.1 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]]
Time = 2.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {x}{2}+\frac {x}{8 \,{\mathrm e}^{x} x^{3}+8 x^{4}+40 \,{\mathrm e}^{x} x^{2}+40 x^{3}+10 \,{\mathrm e}}\) | \(39\) |
| norman | \(\frac {-50 x^{3}+\left (\frac {1}{2}+\frac {5 \,{\mathrm e}}{2}\right ) x -50 \,{\mathrm e}^{x} x^{2}+2 x^{5}+2 \,{\mathrm e}^{x} x^{4}-\frac {25 \,{\mathrm e}}{2}}{4 \,{\mathrm e}^{x} x^{3}+4 x^{4}+20 \,{\mathrm e}^{x} x^{2}+20 x^{3}+5 \,{\mathrm e}}\) | \(70\) |
| parallelrisch | \(\frac {16 x^{5}+16 \,{\mathrm e}^{x} x^{4}-400 x^{3}-400 \,{\mathrm e}^{x} x^{2}+20 x \,{\mathrm e}-100 \,{\mathrm e}+4 x}{32 \,{\mathrm e}^{x} x^{3}+32 x^{4}+160 \,{\mathrm e}^{x} x^{2}+160 x^{3}+40 \,{\mathrm e}}\) | \(71\) |
int(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x^7+320 *x^6+800*x^5-4*x^4-28*x^3-20*x^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3+5)*ex p(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^4)*exp(x )^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp(1)^2+( 80*x^4+400*x^3)*exp(1)+32*x^8+320*x^7+800*x^6),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\frac {4 \, x^{5} + 20 \, x^{4} + 5 \, x e + 4 \, {\left (x^{4} + 5 \, x^{3}\right )} e^{x} + x}{2 \, {\left (4 \, x^{4} + 20 \, x^{3} + 4 \, {\left (x^{3} + 5 \, x^{2}\right )} e^{x} + 5 \, e\right )}} \]
integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x ^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3 +5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^4) *exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp( 1)^2+(80*x^4+400*x^3)*exp(1)+32*x^8+320*x^7+800*x^6),x, algorithm=\
1/2*(4*x^5 + 20*x^4 + 5*x*e + 4*(x^4 + 5*x^3)*e^x + x)/(4*x^4 + 20*x^3 + 4 *(x^3 + 5*x^2)*e^x + 5*e)
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\frac {x}{2} + \frac {x}{8 x^{4} + 40 x^{3} + \left (8 x^{3} + 40 x^{2}\right ) e^{x} + 10 e} \]
integrate(((16*x**6+160*x**5+400*x**4)*exp(x)**2+((40*x**3+200*x**2)*exp(1 )+32*x**7+320*x**6+800*x**5-4*x**4-28*x**3-20*x**2)*exp(x)+25*exp(1)**2+(4 0*x**4+200*x**3+5)*exp(1)+16*x**8+160*x**7+400*x**6-12*x**4-40*x**3)/((32* x**6+320*x**5+800*x**4)*exp(x)**2+((80*x**3+400*x**2)*exp(1)+64*x**7+640*x **6+1600*x**5)*exp(x)+50*exp(1)**2+(80*x**4+400*x**3)*exp(1)+32*x**8+320*x **7+800*x**6),x)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\frac {4 \, x^{5} + 20 \, x^{4} + x {\left (5 \, e + 1\right )} + 4 \, {\left (x^{4} + 5 \, x^{3}\right )} e^{x}}{2 \, {\left (4 \, x^{4} + 20 \, x^{3} + 4 \, {\left (x^{3} + 5 \, x^{2}\right )} e^{x} + 5 \, e\right )}} \]
integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x ^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3 +5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^4) *exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp( 1)^2+(80*x^4+400*x^3)*exp(1)+32*x^8+320*x^7+800*x^6),x, algorithm=\
1/2*(4*x^5 + 20*x^4 + x*(5*e + 1) + 4*(x^4 + 5*x^3)*e^x)/(4*x^4 + 20*x^3 + 4*(x^3 + 5*x^2)*e^x + 5*e)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\frac {4 \, x^{5} + 4 \, x^{4} e^{x} + 20 \, x^{4} + 20 \, x^{3} e^{x} + 5 \, x e + x}{2 \, {\left (4 \, x^{4} + 4 \, x^{3} e^{x} + 20 \, x^{3} + 20 \, x^{2} e^{x} + 5 \, e\right )}} \]
integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x ^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3 +5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^4) *exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp( 1)^2+(80*x^4+400*x^3)*exp(1)+32*x^8+320*x^7+800*x^6),x, algorithm=\
1/2*(4*x^5 + 4*x^4*e^x + 20*x^4 + 20*x^3*e^x + 5*x*e + x)/(4*x^4 + 4*x^3*e ^x + 20*x^3 + 20*x^2*e^x + 5*e)
Timed out. \[ \int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx=\int \frac {25\,{\mathrm {e}}^2+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (40\,x^3+200\,x^2\right )-20\,x^2-28\,x^3-4\,x^4+800\,x^5+320\,x^6+32\,x^7\right )+\mathrm {e}\,\left (40\,x^4+200\,x^3+5\right )+{\mathrm {e}}^{2\,x}\,\left (16\,x^6+160\,x^5+400\,x^4\right )-40\,x^3-12\,x^4+400\,x^6+160\,x^7+16\,x^8}{50\,{\mathrm {e}}^2+\mathrm {e}\,\left (80\,x^4+400\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (32\,x^6+320\,x^5+800\,x^4\right )+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (80\,x^3+400\,x^2\right )+1600\,x^5+640\,x^6+64\,x^7\right )+800\,x^6+320\,x^7+32\,x^8} \,d x \]
int((25*exp(2) + exp(x)*(exp(1)*(200*x^2 + 40*x^3) - 20*x^2 - 28*x^3 - 4*x ^4 + 800*x^5 + 320*x^6 + 32*x^7) + exp(1)*(200*x^3 + 40*x^4 + 5) + exp(2*x )*(400*x^4 + 160*x^5 + 16*x^6) - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16* x^8)/(50*exp(2) + exp(1)*(400*x^3 + 80*x^4) + exp(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + exp(x)*(exp(1)*(400*x^2 + 80*x^3) + 1600*x^5 + 640*x^6 + 64*x^ 7) + 800*x^6 + 320*x^7 + 32*x^8),x)
int((25*exp(2) + exp(x)*(exp(1)*(200*x^2 + 40*x^3) - 20*x^2 - 28*x^3 - 4*x ^4 + 800*x^5 + 320*x^6 + 32*x^7) + exp(1)*(200*x^3 + 40*x^4 + 5) + exp(2*x )*(400*x^4 + 160*x^5 + 16*x^6) - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16* x^8)/(50*exp(2) + exp(1)*(400*x^3 + 80*x^4) + exp(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + exp(x)*(exp(1)*(400*x^2 + 80*x^3) + 1600*x^5 + 640*x^6 + 64*x^ 7) + 800*x^6 + 320*x^7 + 32*x^8), x)