Integrand size = 107, antiderivative size = 26 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=-4+(5+x) \left (2 x+e^{-\frac {1}{5} x^3 (4+x)^4} x\right ) \] Output:
(2*x+x/exp(1/5*(4+x)^4*x^3))*(5+x)-4
Time = 11.88 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=e^{-\frac {1}{5} x^3 (4+x)^4} \left (1+2 e^{\frac {1}{5} x^3 (4+x)^4}\right ) x (5+x) \] Input:
Integrate[(E^((-256*x^3 - 256*x^4 - 96*x^5 - 16*x^6 - x^7)/5)*(25 + 10*x - 3840*x^3 - 5888*x^4 - 3424*x^5 - 960*x^6 - 131*x^7 - 7*x^8 + E^((256*x^3 + 256*x^4 + 96*x^5 + 16*x^6 + x^7)/5)*(50 + 20*x)))/5,x]
Output:
((1 + 2*E^((x^3*(4 + x)^4)/5))*x*(5 + x))/E^((x^3*(4 + x)^4)/5)
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.
| \(\displaystyle \int \frac {1}{5} \left (-7 x^8-131 x^7-960 x^6-3424 x^5-5888 x^4-3840 x^3+e^{\frac {1}{5} \left (x^7+16 x^6+96 x^5+256 x^4+256 x^3\right )} (20 x+50)+10 x+25\right ) \exp \left (\frac {1}{5} \left (-x^7-16 x^6-96 x^5-256 x^4-256 x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int e^{\frac {1}{5} \left (-x^7-16 x^6-96 x^5-256 x^4-256 x^3\right )} \left (-7 x^8-131 x^7-960 x^6-3424 x^5-5888 x^4-3840 x^3+10 x+10 e^{\frac {1}{5} \left (x^7+16 x^6+96 x^5+256 x^4+256 x^3\right )} (2 x+5)+25\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{5} \int e^{-\frac {1}{5} x^3 (x+4)^4} \left (-7 x^8-131 x^7-960 x^6-3424 x^5-5888 x^4-3840 x^3+10 x+10 e^{\frac {1}{5} \left (x^7+16 x^6+96 x^5+256 x^4+256 x^3\right )} (2 x+5)+25\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (-7 e^{-\frac {1}{5} x^3 (x+4)^4} x^8-131 e^{-\frac {1}{5} x^3 (x+4)^4} x^7-960 e^{-\frac {1}{5} x^3 (x+4)^4} x^6-3424 e^{-\frac {1}{5} x^3 (x+4)^4} x^5-5888 e^{-\frac {1}{5} x^3 (x+4)^4} x^4-3840 e^{-\frac {1}{5} x^3 (x+4)^4} x^3+10 e^{-\frac {1}{5} x^3 (x+4)^4} x+25 e^{-\frac {1}{5} x^3 (x+4)^4}+10 (2 x+5)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (25 \int e^{-\frac {1}{5} x^3 (x+4)^4}dx+10 \int e^{-\frac {1}{5} x^3 (x+4)^4} xdx-3840 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^3dx-7 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^8dx-131 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^7dx-960 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^6dx-3424 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^5dx-5888 \int e^{-\frac {1}{5} x^3 (x+4)^4} x^4dx+\frac {5}{2} (2 x+5)^2\right )\) |
Input:
Int[(E^((-256*x^3 - 256*x^4 - 96*x^5 - 16*x^6 - x^7)/5)*(25 + 10*x - 3840* x^3 - 5888*x^4 - 3424*x^5 - 960*x^6 - 131*x^7 - 7*x^8 + E^((256*x^3 + 256* x^4 + 96*x^5 + 16*x^6 + x^7)/5)*(50 + 20*x)))/5,x]
Output:
$Aborted
Time = 3.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(2 x^{2}+10 x +\frac {\left (5 x^{2}+25 x \right ) {\mathrm e}^{-\frac {\left (4+x \right )^{4} x^{3}}{5}}}{5}\) | \(32\) |
| parts | \(10 x +\left (x^{2}+5 x \right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}}+2 x^{2}\) | \(47\) |
| parallelrisch | \(\frac {\left (50 x^{2} {\mathrm e}^{\frac {x^{3} \left (x^{4}+16 x^{3}+96 x^{2}+256 x +256\right )}{5}}+25 x^{2}+250 \,{\mathrm e}^{\frac {x^{3} \left (x^{4}+16 x^{3}+96 x^{2}+256 x +256\right )}{5}} x +125 x \right ) {\mathrm e}^{-\frac {x^{3} \left (x^{4}+16 x^{3}+96 x^{2}+256 x +256\right )}{5}}}{25}\) | \(94\) |
| norman | \(\left (x^{2}+5 x +2 x^{2} {\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}+10 \,{\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}} x \right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}}\) | \(100\) |
| orering | \(\frac {\left (343 x^{16}+12838 x^{15}+135072 x^{14}-470043 x^{13}-23281600 x^{12}-243480768 x^{11}-1398693888 x^{10}-5026808722 x^{9}-11586210882 x^{8}-16644310630 x^{7}-13543000440 x^{6}-4709520480 x^{5}+168337600 x^{4}+241241600 x^{3}+124032000 x^{2}-80750\right ) \left (\left (20 x +50\right ) {\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}-7 x^{8}-131 x^{7}-960 x^{6}-3424 x^{5}-5888 x^{4}-3840 x^{3}+10 x +25\right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}}}{35 x^{2} \left (98 x^{11}+2639 x^{10}+30577 x^{9}+198792 x^{8}+791792 x^{7}+1974784 x^{6}+3006208 x^{5}+2549130 x^{4}+914760 x^{3}-26200 x^{2}-42400 x -24000\right ) \left (4+x \right )^{2}}+\frac {5 \left (49 x^{10}+1162 x^{9}-153997 x^{7}-1389344 x^{6}-5296800 x^{5}-9374720 x^{4}-6201600 x^{3}+16150 x +40375\right ) \left (\frac {\left (20 \,{\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}+\left (20 x +50\right ) \left (\frac {7}{5} x^{6}+\frac {96}{5} x^{5}+96 x^{4}+\frac {1024}{5} x^{3}+\frac {768}{5} x^{2}\right ) {\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}-56 x^{7}-917 x^{6}-5760 x^{5}-17120 x^{4}-23552 x^{3}-11520 x^{2}+10\right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}}}{5}-\frac {\left (\left (20 x +50\right ) {\mathrm e}^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}-7 x^{8}-131 x^{7}-960 x^{6}-3424 x^{5}-5888 x^{4}-3840 x^{3}+10 x +25\right ) {\mathrm e}^{-\frac {1}{5} x^{7}-\frac {16}{5} x^{6}-\frac {96}{5} x^{5}-\frac {256}{5} x^{4}-\frac {256}{5} x^{3}} \left (\frac {7}{5} x^{6}+\frac {96}{5} x^{5}+96 x^{4}+\frac {1024}{5} x^{3}+\frac {768}{5} x^{2}\right )}{5}\right )}{7 x^{2} \left (98 x^{11}+2639 x^{10}+30577 x^{9}+198792 x^{8}+791792 x^{7}+1974784 x^{6}+3006208 x^{5}+2549130 x^{4}+914760 x^{3}-26200 x^{2}-42400 x -24000\right ) \left (4+x \right )^{2}}\) | \(627\) |
Input:
int(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^ 8-131*x^7-960*x^6-3424*x^5-5888*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6 +96/5*x^5+256/5*x^4+256/5*x^3),x,method=_RETURNVERBOSE)
Output:
2*x^2+10*x+1/5*(5*x^2+25*x)*exp(-1/5*(4+x)^4*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx={\left (x^{2} + 2 \, {\left (x^{2} + 5 \, x\right )} e^{\left (\frac {1}{5} \, x^{7} + \frac {16}{5} \, x^{6} + \frac {96}{5} \, x^{5} + \frac {256}{5} \, x^{4} + \frac {256}{5} \, x^{3}\right )} + 5 \, x\right )} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} \] Input:
integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3 )-7*x^8-131*x^7-960*x^6-3424*x^5-5888*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16 /5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="fricas")
Output:
(x^2 + 2*(x^2 + 5*x)*e^(1/5*x^7 + 16/5*x^6 + 96/5*x^5 + 256/5*x^4 + 256/5* x^3) + 5*x)*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=2 x^{2} + 10 x + \left (x^{2} + 5 x\right ) e^{- \frac {x^{7}}{5} - \frac {16 x^{6}}{5} - \frac {96 x^{5}}{5} - \frac {256 x^{4}}{5} - \frac {256 x^{3}}{5}} \] Input:
integrate(1/5*((20*x+50)*exp(1/5*x**7+16/5*x**6+96/5*x**5+256/5*x**4+256/5 *x**3)-7*x**8-131*x**7-960*x**6-3424*x**5-5888*x**4-3840*x**3+10*x+25)/exp (1/5*x**7+16/5*x**6+96/5*x**5+256/5*x**4+256/5*x**3),x)
Output:
2*x**2 + 10*x + (x**2 + 5*x)*exp(-x**7/5 - 16*x**6/5 - 96*x**5/5 - 256*x** 4/5 - 256*x**3/5)
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=2 \, x^{2} + {\left (x^{2} + 5 \, x\right )} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 10 \, x \] Input:
integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3 )-7*x^8-131*x^7-960*x^6-3424*x^5-5888*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16 /5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="maxima")
Output:
2*x^2 + (x^2 + 5*x)*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5* x^3) + 10*x
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (23) = 46\).
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=x^{2} e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 2 \, x^{2} + 5 \, x e^{\left (-\frac {1}{5} \, x^{7} - \frac {16}{5} \, x^{6} - \frac {96}{5} \, x^{5} - \frac {256}{5} \, x^{4} - \frac {256}{5} \, x^{3}\right )} + 10 \, x \] Input:
integrate(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3 )-7*x^8-131*x^7-960*x^6-3424*x^5-5888*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16 /5*x^6+96/5*x^5+256/5*x^4+256/5*x^3),x, algorithm="giac")
Output:
x^2*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5*x^3) + 2*x^2 + 5 *x*e^(-1/5*x^7 - 16/5*x^6 - 96/5*x^5 - 256/5*x^4 - 256/5*x^3) + 10*x
Time = 2.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=x\,\left ({\mathrm {e}}^{-\frac {x^7}{5}-\frac {16\,x^6}{5}-\frac {96\,x^5}{5}-\frac {256\,x^4}{5}-\frac {256\,x^3}{5}}+2\right )\,\left (x+5\right ) \] Input:
int(-exp(- (256*x^3)/5 - (256*x^4)/5 - (96*x^5)/5 - (16*x^6)/5 - x^7/5)*(7 68*x^3 - (exp((256*x^3)/5 + (256*x^4)/5 + (96*x^5)/5 + (16*x^6)/5 + x^7/5) *(20*x + 50))/5 - 2*x + (5888*x^4)/5 + (3424*x^5)/5 + 192*x^6 + (131*x^7)/ 5 + (7*x^8)/5 - 5),x)
Output:
x*(exp(- (256*x^3)/5 - (256*x^4)/5 - (96*x^5)/5 - (16*x^6)/5 - x^7/5) + 2) *(x + 5)
Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.69 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-256 x^3-256 x^4-96 x^5-16 x^6-x^7\right )} \left (25+10 x-3840 x^3-5888 x^4-3424 x^5-960 x^6-131 x^7-7 x^8+e^{\frac {1}{5} \left (256 x^3+256 x^4+96 x^5+16 x^6+x^7\right )} (50+20 x)\right ) \, dx=\frac {x \left (2 e^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}} x +10 e^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}+x +5\right )}{e^{\frac {1}{5} x^{7}+\frac {16}{5} x^{6}+\frac {96}{5} x^{5}+\frac {256}{5} x^{4}+\frac {256}{5} x^{3}}} \] Input:
int(1/5*((20*x+50)*exp(1/5*x^7+16/5*x^6+96/5*x^5+256/5*x^4+256/5*x^3)-7*x^ 8-131*x^7-960*x^6-3424*x^5-5888*x^4-3840*x^3+10*x+25)/exp(1/5*x^7+16/5*x^6 +96/5*x^5+256/5*x^4+256/5*x^3),x)
Output:
(x*(2*e**((x**7 + 16*x**6 + 96*x**5 + 256*x**4 + 256*x**3)/5)*x + 10*e**(( x**7 + 16*x**6 + 96*x**5 + 256*x**4 + 256*x**3)/5) + x + 5))/e**((x**7 + 1 6*x**6 + 96*x**5 + 256*x**4 + 256*x**3)/5)