\(\int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7))}{5 x^5+40 x^6+80 x^7} \, dx\) [441]

Optimal result

Integrand size = 105, antiderivative size = 28 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=-\frac {1}{x^4}+x+e^{\frac {1}{5} \left (x+\frac {x}{1+4 x}\right )} x^2 \] Output:

x-1/x^4+x^2/exp(-1/5*x-1/5*x/(1+4*x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=\frac {-1+x^5+e^{\frac {2 x (1+2 x)}{5+20 x}} x^6}{x^4} \] Input:

Integrate[(10*x^6 + 82*x^7 + 168*x^8 + 16*x^9 + E^((-2*x - 4*x^2)/(5 + 20* 
x))*(20 + 160*x + 320*x^2 + 5*x^5 + 40*x^6 + 80*x^7))/(E^((-2*x - 4*x^2)/( 
5 + 20*x))*(5*x^5 + 40*x^6 + 80*x^7)),x]
 

Output:

(-1 + x^5 + E^((2*x*(1 + 2*x))/(5 + 20*x))*x^6)/x^4
 

Rubi [F]

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.

\(\displaystyle \int \frac {e^{-\frac {-4 x^2-2 x}{20 x+5}} \left (16 x^9+168 x^8+82 x^7+10 x^6+e^{\frac {-4 x^2-2 x}{20 x+5}} \left (80 x^7+40 x^6+5 x^5+320 x^2+160 x+20\right )\right )}{80 x^7+40 x^6+5 x^5} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{-\frac {-4 x^2-2 x}{20 x+5}} \left (16 x^9+168 x^8+82 x^7+10 x^6+e^{\frac {-4 x^2-2 x}{20 x+5}} \left (80 x^7+40 x^6+5 x^5+320 x^2+160 x+20\right )\right )}{x^5 \left (80 x^2+40 x+5\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {e^{-\frac {-4 x^2-2 x}{20 x+5}} \left (16 x^9+168 x^8+82 x^7+10 x^6+e^{\frac {-4 x^2-2 x}{20 x+5}} \left (80 x^7+40 x^6+5 x^5+320 x^2+160 x+20\right )\right )}{x^5 \left (4 \sqrt {5} x+\sqrt {5}\right )^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{-\frac {(-4 x-2) x}{20 x+5}} \left (16 x^9+168 x^8+82 x^7+10 x^6+e^{\frac {-4 x^2-2 x}{20 x+5}} \left (80 x^7+40 x^6+5 x^5+320 x^2+160 x+20\right )\right )}{x^5 \left (4 \sqrt {5} x+\sqrt {5}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (x^5+4\right ) \exp \left (-\frac {(-4 x-2) x}{20 x+5}-\frac {2 (2 x+1) x}{20 x+5}\right )}{x^5}+\frac {16 e^{-\frac {(-4 x-2) x}{20 x+5}} x^4}{5 (4 x+1)^2}+\frac {168 e^{-\frac {(-4 x-2) x}{20 x+5}} x^3}{5 (4 x+1)^2}+\frac {82 e^{-\frac {(-4 x-2) x}{20 x+5}} x^2}{5 (4 x+1)^2}+\frac {2 e^{-\frac {(-4 x-2) x}{20 x+5}} x}{(4 x+1)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\frac {4}{x^5}+\frac {2 e^{\frac {2 x (2 x+1)}{20 x+5}} x \left (8 x^3+84 x^2+41 x+5\right )}{5 (4 x+1)^2}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} \int e^{\frac {2 x (2 x+1)}{20 x+5}} x^2dx+\frac {1}{80} \int e^{\frac {2 x (2 x+1)}{20 x+5}}dx+2 \int e^{\frac {2 x (2 x+1)}{20 x+5}} xdx+\frac {1}{80} \int \frac {e^{\frac {2 x (2 x+1)}{20 x+5}}}{(4 x+1)^2}dx-\frac {1}{40} \int \frac {e^{\frac {2 x (2 x+1)}{20 x+5}}}{4 x+1}dx-\frac {1}{x^4}+x\)

Input:

Int[(10*x^6 + 82*x^7 + 168*x^8 + 16*x^9 + E^((-2*x - 4*x^2)/(5 + 20*x))*(2 
0 + 160*x + 320*x^2 + 5*x^5 + 40*x^6 + 80*x^7))/(E^((-2*x - 4*x^2)/(5 + 20 
*x))*(5*x^5 + 40*x^6 + 80*x^7)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
risch	\(x -\frac {1}{x^{4}}+x^{2} {\mathrm e}^{\frac {2 \left (1+2 x \right ) x}{5 \left (1+4 x \right )}}\)	\(28\)
parts	\(x -\frac {1}{x^{4}}+\frac {\left (4 x^{3}+x^{2}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}}}{1+4 x}\)	\(45\)
norman	\(\frac {\left (x^{6}-\frac {x^{4} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}}{4}+4 x^{7}-4 x \,{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+4 x^{6} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}-{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}}}{x^{4} \left (1+4 x \right )}\)	\(128\)
parallelrisch	\(\frac {\left (15360 x^{7}+15360 x^{6} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+3840 x^{6}+1280 \,{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}} x^{5}-640 x^{4} {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}-15360 x \,{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}-3840 \,{\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}\right ) {\mathrm e}^{\frac {2 \left (1+2 x \right ) x}{5 \left (1+4 x \right )}}}{3840 x^{4} \left (1+4 x \right )}\)	\(157\)
orering	\(\frac {x \left (256 x^{11}+3968 x^{10}-11520 x^{9}-84352 x^{8}-76772 x^{7}-28258 x^{6}-9946 x^{5}+109677 x^{4}+113088 x^{3}+42756 x^{2}+7160 x +450\right ) \left (\left (80 x^{7}+40 x^{6}+5 x^{5}+320 x^{2}+160 x +20\right ) {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+16 x^{9}+168 x^{8}+82 x^{7}+10 x^{6}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}}}{2 \left (128 x^{11}+2688 x^{10}+9024 x^{9}+7456 x^{8}+2622 x^{7}+932 x^{6}+23577 x^{5}+176896 x^{4}+163424 x^{3}+59688 x^{2}+9780 x +600\right ) \left (80 x^{7}+40 x^{6}+5 x^{5}\right )}-\frac {5 x^{2} \left (16 x^{8}-882 x^{6}-446 x^{5}-55 x^{4}-16 x^{3}-488 x^{2}-242 x -30\right ) \left (1+4 x \right )^{2} \left (\frac {\left (\left (560 x^{6}+240 x^{5}+25 x^{4}+640 x +160\right ) {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+\left (80 x^{7}+40 x^{6}+5 x^{5}+320 x^{2}+160 x +20\right ) \left (\frac {-8 x -2}{20 x +5}-\frac {20 \left (-4 x^{2}-2 x \right )}{\left (20 x +5\right )^{2}}\right ) {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+144 x^{8}+1344 x^{7}+574 x^{6}+60 x^{5}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}}}{80 x^{7}+40 x^{6}+5 x^{5}}-\frac {\left (\left (80 x^{7}+40 x^{6}+5 x^{5}+320 x^{2}+160 x +20\right ) {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+16 x^{9}+168 x^{8}+82 x^{7}+10 x^{6}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}} \left (560 x^{6}+240 x^{5}+25 x^{4}\right )}{\left (80 x^{7}+40 x^{6}+5 x^{5}\right )^{2}}-\frac {\left (\left (80 x^{7}+40 x^{6}+5 x^{5}+320 x^{2}+160 x +20\right ) {\mathrm e}^{\frac {-4 x^{2}-2 x}{20 x +5}}+16 x^{9}+168 x^{8}+82 x^{7}+10 x^{6}\right ) {\mathrm e}^{-\frac {-4 x^{2}-2 x}{20 x +5}} \left (\frac {-8 x -2}{20 x +5}-\frac {20 \left (-4 x^{2}-2 x \right )}{\left (20 x +5\right )^{2}}\right )}{80 x^{7}+40 x^{6}+5 x^{5}}\right )}{2 \left (128 x^{11}+2688 x^{10}+9024 x^{9}+7456 x^{8}+2622 x^{7}+932 x^{6}+23577 x^{5}+176896 x^{4}+163424 x^{3}+59688 x^{2}+9780 x +600\right )}\)	\(758\)

Input:

int(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16* 
x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5) 
),x,method=_RETURNVERBOSE)
 

Output:

x-1/x^4+x^2*exp(2/5*(1+2*x)*x/(1+4*x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=\frac {x^{6} e^{\left (\frac {2 \, {\left (2 \, x^{2} + x\right )}}{5 \, {\left (4 \, x + 1\right )}}\right )} + x^{5} - 1}{x^{4}} \] Input:

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5 
))+16*x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(2 
0*x+5)),x, algorithm="fricas")
 

Output:

(x^6*e^(2/5*(2*x^2 + x)/(4*x + 1)) + x^5 - 1)/x^4
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=x^{2} e^{- \frac {- 4 x^{2} - 2 x}{20 x + 5}} + x - \frac {1}{x^{4}} \] Input:

integrate(((80*x**7+40*x**6+5*x**5+320*x**2+160*x+20)*exp((-4*x**2-2*x)/(2 
0*x+5))+16*x**9+168*x**8+82*x**7+10*x**6)/(80*x**7+40*x**6+5*x**5)/exp((-4 
*x**2-2*x)/(20*x+5)),x)
 

Output:

x**2*exp(-(-4*x**2 - 2*x)/(20*x + 5)) + x - 1/x**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.75 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=x^{2} e^{\left (\frac {1}{5} \, x - \frac {1}{20 \, {\left (4 \, x + 1\right )}} + \frac {1}{20}\right )} + x + \frac {15360 \, x^{4} + 1920 \, x^{3} - 160 \, x^{2} + 20 \, x - 3}{3 \, {\left (4 \, x^{5} + x^{4}\right )}} - \frac {32 \, {\left (768 \, x^{3} + 96 \, x^{2} - 8 \, x + 1\right )}}{3 \, {\left (4 \, x^{4} + x^{3}\right )}} + \frac {32 \, {\left (96 \, x^{2} + 12 \, x - 1\right )}}{4 \, x^{3} + x^{2}} \] Input:

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5 
))+16*x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(2 
0*x+5)),x, algorithm="maxima")
 

Output:

x^2*e^(1/5*x - 1/20/(4*x + 1) + 1/20) + x + 1/3*(15360*x^4 + 1920*x^3 - 16 
0*x^2 + 20*x - 3)/(4*x^5 + x^4) - 32/3*(768*x^3 + 96*x^2 - 8*x + 1)/(4*x^4 
 + x^3) + 32*(96*x^2 + 12*x - 1)/(4*x^3 + x^2)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=\frac {x^{6} e^{\left (\frac {2 \, {\left (2 \, x^{2} + x\right )}}{5 \, {\left (4 \, x + 1\right )}}\right )} + x^{5} - 1}{x^{4}} \] Input:

integrate(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5 
))+16*x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(2 
0*x+5)),x, algorithm="giac")
 

Output:

(x^6*e^(2/5*(2*x^2 + x)/(4*x + 1)) + x^5 - 1)/x^4
 

Mupad [B] (verification not implemented)

Time = 2.69 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=x-\frac {1}{x^4}+x^2\,{\mathrm {e}}^{\frac {4\,x^2}{20\,x+5}}\,{\mathrm {e}}^{\frac {2\,x}{20\,x+5}} \] Input:

int((exp((2*x + 4*x^2)/(20*x + 5))*(exp(-(2*x + 4*x^2)/(20*x + 5))*(160*x 
+ 320*x^2 + 5*x^5 + 40*x^6 + 80*x^7 + 20) + 10*x^6 + 82*x^7 + 168*x^8 + 16 
*x^9))/(5*x^5 + 40*x^6 + 80*x^7),x)
 

Output:

x - 1/x^4 + x^2*exp((4*x^2)/(20*x + 5))*exp((2*x)/(20*x + 5))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {-2 x-4 x^2}{5+20 x}} \left (10 x^6+82 x^7+168 x^8+16 x^9+e^{\frac {-2 x-4 x^2}{5+20 x}} \left (20+160 x+320 x^2+5 x^5+40 x^6+80 x^7\right )\right )}{5 x^5+40 x^6+80 x^7} \, dx=\frac {e^{\frac {4 x^{2}+2 x}{20 x +5}} x^{6}+x^{5}-1}{x^{4}} \] Input:

int(((80*x^7+40*x^6+5*x^5+320*x^2+160*x+20)*exp((-4*x^2-2*x)/(20*x+5))+16* 
x^9+168*x^8+82*x^7+10*x^6)/(80*x^7+40*x^6+5*x^5)/exp((-4*x^2-2*x)/(20*x+5) 
),x)
 

Output:

(e**((4*x**2 + 2*x)/(20*x + 5))*x**6 + x**5 - 1)/x**4