Integrand size = 117, antiderivative size = 18 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\left (47+\frac {e^x}{\left (1-\frac {1}{x}+x\right )^2}\right )^2 \] Output:
(exp(x)/(1+x-1/x)^2+47)^2
Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).
Time = 3.53 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {e^x x^2 \left (94-188 x+\left (-94+e^x\right ) x^2+188 x^3+94 x^4\right )}{\left (-1+x+x^2\right )^4} \] Input:
Integrate[(E^(2*x)*(-4*x^3 - 2*x^4 - 2*x^5 + 2*x^6) + E^x*(-188*x + 282*x^ 2 + 282*x^3 - 470*x^5 - 376*x^6 + 94*x^7 + 94*x^8))/(-1 + 5*x - 5*x^2 - 10 *x^3 + 15*x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10),x]
Output:
(E^x*x^2*(94 - 188*x + (-94 + E^x)*x^2 + 188*x^3 + 94*x^4))/(-1 + x + x^2) ^4
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.
| \(\displaystyle \int \frac {e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )}{x^{10}+5 x^9+5 x^8-10 x^7-15 x^6+11 x^5+15 x^4-10 x^3-5 x^2+5 x-1} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {28 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{125 \sqrt {5} \left (-2 x+\sqrt {5}-1\right )}-\frac {28 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{125 \sqrt {5} \left (2 x+\sqrt {5}+1\right )}-\frac {28 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{125 \left (-2 x+\sqrt {5}-1\right )^2}-\frac {28 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{125 \left (2 x+\sqrt {5}+1\right )^2}-\frac {24 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \sqrt {5} \left (-2 x+\sqrt {5}-1\right )^3}-\frac {24 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \sqrt {5} \left (2 x+\sqrt {5}+1\right )^3}-\frac {16 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \left (-2 x+\sqrt {5}-1\right )^4}-\frac {16 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \left (2 x+\sqrt {5}+1\right )^4}-\frac {32 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \sqrt {5} \left (-2 x+\sqrt {5}-1\right )^5}-\frac {32 \left (e^{2 x} \left (2 x^6-2 x^5-2 x^4-4 x^3\right )+e^x \left (94 x^8+94 x^7-376 x^6-470 x^5+282 x^3+282 x^2-188 x\right )\right )}{25 \sqrt {5} \left (2 x+\sqrt {5}+1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (e^x-47\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3-\left (47-e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^x x \left (-x^3+x^2+x+2\right ) \left (47 x^4+94 x^3+\left (-47+e^x\right ) x^2-94 x+47\right )}{\left (-x^2-x+1\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^5}{\left (x^2+x-1\right )^5}+\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^4}{\left (x^2+x-1\right )^5}-\frac {47 e^x (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}+\frac {e^{2 x} (x-2) \left (x^2+x+1\right ) x^3}{\left (x^2+x-1\right )^5}-\frac {94 e^x (x-2) \left (x^2+x+1\right ) x^2}{\left (x^2+x-1\right )^5}+\frac {47 e^x (x-2) \left (x^2+x+1\right ) x}{\left (x^2+x-1\right )^5}\right )dx\) |
Input:
Int[(E^(2*x)*(-4*x^3 - 2*x^4 - 2*x^5 + 2*x^6) + E^x*(-188*x + 282*x^2 + 28 2*x^3 - 470*x^5 - 376*x^6 + 94*x^7 + 94*x^8))/(-1 + 5*x - 5*x^2 - 10*x^3 + 15*x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10),x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83
| method | result | size |
| risch | \(\frac {x^{4} {\mathrm e}^{2 x}}{\left (x^{2}+x -1\right )^{4}}+\frac {94 x^{2} {\mathrm e}^{x}}{\left (x^{2}+x -1\right )^{2}}\) | \(33\) |
| norman | \(\frac {{\mathrm e}^{2 x} x^{4}+188 x^{5} {\mathrm e}^{x}+94 x^{6} {\mathrm e}^{x}+94 \,{\mathrm e}^{x} x^{2}-188 \,{\mathrm e}^{x} x^{3}-94 \,{\mathrm e}^{x} x^{4}}{\left (x^{2}+x -1\right )^{4}}\) | \(54\) |
| parallelrisch | \(\frac {188 x^{6} {\mathrm e}^{x}+376 x^{5} {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} x^{4}-188 \,{\mathrm e}^{x} x^{4}-376 \,{\mathrm e}^{x} x^{3}+188 \,{\mathrm e}^{x} x^{2}}{2 x^{8}+8 x^{7}+4 x^{6}-16 x^{5}-10 x^{4}+16 x^{3}+4 x^{2}-8 x +2}\) | \(88\) |
| parts | \(-\frac {{\mathrm e}^{2 x} \left (128 x^{7}+538 x^{6}+552 x^{5}-870 x^{4}-340 x^{3}+574 x^{2}-126 x -9\right )}{750 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {{\mathrm e}^{2 x} \left (12 x^{7}+2 x^{6}-142 x^{5}-205 x^{4}+240 x^{3}+96 x^{2}-154 x +39\right )}{250 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {{\mathrm e}^{2 x} \left (56 x^{7}+176 x^{6}+4 x^{5}-290 x^{4}+120 x^{3}+448 x^{2}-302 x +57\right )}{375 x^{8}+1500 x^{7}+750 x^{6}-3000 x^{5}-1875 x^{4}+3000 x^{3}+750 x^{2}-1500 x +375}+\frac {{\mathrm e}^{2 x} \left (52 x^{7}+192 x^{6}+118 x^{5}-155 x^{4}+140 x^{3}-34 x^{2}+16 x -6\right )}{750 x^{8}+3000 x^{7}+1500 x^{6}-6000 x^{5}-3750 x^{4}+6000 x^{3}+1500 x^{2}-3000 x +750}-\frac {47 \,{\mathrm e}^{x} \left (23 x^{3}+2 x^{2}-9 x +1\right )}{25 \left (x^{4}+2 x^{3}-x^{2}-2 x +1\right )}+\frac {94 \,{\mathrm e}^{x} \left (11 x^{3}+14 x^{2}-13 x +7\right )}{25 \left (x^{4}+2 x^{3}-x^{2}-2 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (3 x^{3}-3 x^{2}-24 x +11\right )}{25 \left (x^{4}+2 x^{3}-x^{2}-2 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (4 x^{3}+21 x^{2}-7 x -2\right )}{25 \left (x^{4}+2 x^{3}-x^{2}-2 x +1\right )}\) | \(482\) |
| orering | \(\frac {x \left (x^{2}+x -1\right ) \left (3 x^{6}-6 x^{5}-x^{4}-6 x^{3}+23 x^{2}+16 x +10\right ) \left (\left (2 x^{6}-2 x^{5}-2 x^{4}-4 x^{3}\right ) {\mathrm e}^{2 x}+\left (94 x^{8}+94 x^{7}-376 x^{6}-470 x^{5}+282 x^{3}+282 x^{2}-188 x \right ) {\mathrm e}^{x}\right )}{2 \left (-2+x \right )^{3} \left (x^{2}+x +1\right )^{3} \left (x^{10}+5 x^{9}+5 x^{8}-10 x^{7}-15 x^{6}+11 x^{5}+15 x^{4}-10 x^{3}-5 x^{2}+5 x -1\right )}-\frac {x^{2} \left (x^{2}+x -1\right )^{2} \left (\frac {\left (12 x^{5}-10 x^{4}-8 x^{3}-12 x^{2}\right ) {\mathrm e}^{2 x}+2 \left (2 x^{6}-2 x^{5}-2 x^{4}-4 x^{3}\right ) {\mathrm e}^{2 x}+\left (752 x^{7}+658 x^{6}-2256 x^{5}-2350 x^{4}+846 x^{2}+564 x -188\right ) {\mathrm e}^{x}+\left (94 x^{8}+94 x^{7}-376 x^{6}-470 x^{5}+282 x^{3}+282 x^{2}-188 x \right ) {\mathrm e}^{x}}{x^{10}+5 x^{9}+5 x^{8}-10 x^{7}-15 x^{6}+11 x^{5}+15 x^{4}-10 x^{3}-5 x^{2}+5 x -1}-\frac {\left (\left (2 x^{6}-2 x^{5}-2 x^{4}-4 x^{3}\right ) {\mathrm e}^{2 x}+\left (94 x^{8}+94 x^{7}-376 x^{6}-470 x^{5}+282 x^{3}+282 x^{2}-188 x \right ) {\mathrm e}^{x}\right ) \left (10 x^{9}+45 x^{8}+40 x^{7}-70 x^{6}-90 x^{5}+55 x^{4}+60 x^{3}-30 x^{2}-10 x +5\right )}{\left (x^{10}+5 x^{9}+5 x^{8}-10 x^{7}-15 x^{6}+11 x^{5}+15 x^{4}-10 x^{3}-5 x^{2}+5 x -1\right )^{2}}\right )}{2 \left (-2+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\) | \(531\) |
| default | \(-\frac {47 \,{\mathrm e}^{x} \left (7 x^{7}-53 x^{6}-377 x^{5}-870 x^{4}+685 x^{3}+446 x^{2}-514 x +119\right )}{300 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {{\mathrm e}^{2 x} \left (12 x^{7}+2 x^{6}-142 x^{5}-205 x^{4}+240 x^{3}+96 x^{2}-154 x +39\right )}{250 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (4 x^{7}-241 x^{6}-1269 x^{5}+185 x^{4}+1145 x^{3}-263 x^{2}-333 x +118\right )}{375 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (253 x^{7}+788 x^{6}-108 x^{5}-1705 x^{4}+140 x^{3}+1459 x^{2}-531 x +151\right )}{750 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (109 x^{7}+339 x^{6}-49 x^{5}-740 x^{4}+45 x^{3}+552 x^{2}-668 x +153\right )}{500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {47 \,{\mathrm e}^{x} \left (18 x^{7}+53 x^{6}-23 x^{5}-155 x^{4}-85 x^{3}-371 x^{2}+239 x -44\right )}{500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (437 x^{7}+1952 x^{6}-232 x^{5}-2445 x^{4}+560 x^{3}+1111 x^{2}-599 x +79\right )}{1500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}-\frac {47 \,{\mathrm e}^{x} \left (399 x^{7}-971 x^{6}+761 x^{5}+2960 x^{4}-2405 x^{3}-1828 x^{2}+2152 x -517\right )}{1500 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}+\frac {{\mathrm e}^{2 x} \left (56 x^{7}+176 x^{6}+4 x^{5}-290 x^{4}+120 x^{3}+448 x^{2}-302 x +57\right )}{375 x^{8}+1500 x^{7}+750 x^{6}-3000 x^{5}-1875 x^{4}+3000 x^{3}+750 x^{2}-1500 x +375}+\frac {{\mathrm e}^{2 x} \left (52 x^{7}+192 x^{6}+118 x^{5}-155 x^{4}+140 x^{3}-34 x^{2}+16 x -6\right )}{750 x^{8}+3000 x^{7}+1500 x^{6}-6000 x^{5}-3750 x^{4}+6000 x^{3}+1500 x^{2}-3000 x +750}-\frac {{\mathrm e}^{2 x} \left (128 x^{7}+538 x^{6}+552 x^{5}-870 x^{4}-340 x^{3}+574 x^{2}-126 x -9\right )}{750 \left (x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1\right )}\) | \(879\) |
Input:
int(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282 *x^3+282*x^2-188*x)*exp(x))/(x^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4- 10*x^3-5*x^2+5*x-1),x,method=_RETURNVERBOSE)
Output:
x^4/(x^2+x-1)^4*exp(x)^2+94*x^2/(x^2+x-1)^2*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.22 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} + 94 \, {\left (x^{6} + 2 \, x^{5} - x^{4} - 2 \, x^{3} + x^{2}\right )} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \] Input:
integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x ^5+282*x^3+282*x^2-188*x)*exp(x))/(x^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+1 5*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="fricas")
Output:
(x^4*e^(2*x) + 94*(x^6 + 2*x^5 - x^4 - 2*x^3 + x^2)*e^x)/(x^8 + 4*x^7 + 2* x^6 - 8*x^5 - 5*x^4 + 8*x^3 + 2*x^2 - 4*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (14) = 28\).
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 7.39 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {\left (x^{8} + 2 x^{7} - x^{6} - 2 x^{5} + x^{4}\right ) e^{2 x} + \left (94 x^{10} + 376 x^{9} + 188 x^{8} - 752 x^{7} - 470 x^{6} + 752 x^{5} + 188 x^{4} - 376 x^{3} + 94 x^{2}\right ) e^{x}}{x^{12} + 6 x^{11} + 9 x^{10} - 10 x^{9} - 30 x^{8} + 6 x^{7} + 41 x^{6} - 6 x^{5} - 30 x^{4} + 10 x^{3} + 9 x^{2} - 6 x + 1} \] Input:
integrate(((2*x**6-2*x**5-2*x**4-4*x**3)*exp(x)**2+(94*x**8+94*x**7-376*x* *6-470*x**5+282*x**3+282*x**2-188*x)*exp(x))/(x**10+5*x**9+5*x**8-10*x**7- 15*x**6+11*x**5+15*x**4-10*x**3-5*x**2+5*x-1),x)
Output:
((x**8 + 2*x**7 - x**6 - 2*x**5 + x**4)*exp(2*x) + (94*x**10 + 376*x**9 + 188*x**8 - 752*x**7 - 470*x**6 + 752*x**5 + 188*x**4 - 376*x**3 + 94*x**2) *exp(x))/(x**12 + 6*x**11 + 9*x**10 - 10*x**9 - 30*x**8 + 6*x**7 + 41*x**6 - 6*x**5 - 30*x**4 + 10*x**3 + 9*x**2 - 6*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.22 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} + 94 \, {\left (x^{6} + 2 \, x^{5} - x^{4} - 2 \, x^{3} + x^{2}\right )} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \] Input:
integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x ^5+282*x^3+282*x^2-188*x)*exp(x))/(x^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+1 5*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="maxima")
Output:
(x^4*e^(2*x) + 94*(x^6 + 2*x^5 - x^4 - 2*x^3 + x^2)*e^x)/(x^8 + 4*x^7 + 2* x^6 - 8*x^5 - 5*x^4 + 8*x^3 + 2*x^2 - 4*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (17) = 34\).
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.72 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {94 \, x^{6} e^{x} + 188 \, x^{5} e^{x} + x^{4} e^{\left (2 \, x\right )} - 94 \, x^{4} e^{x} - 188 \, x^{3} e^{x} + 94 \, x^{2} e^{x}}{x^{8} + 4 \, x^{7} + 2 \, x^{6} - 8 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} - 4 \, x + 1} \] Input:
integrate(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x ^5+282*x^3+282*x^2-188*x)*exp(x))/(x^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+1 5*x^4-10*x^3-5*x^2+5*x-1),x, algorithm="giac")
Output:
(94*x^6*e^x + 188*x^5*e^x + x^4*e^(2*x) - 94*x^4*e^x - 188*x^3*e^x + 94*x^ 2*e^x)/(x^8 + 4*x^7 + 2*x^6 - 8*x^5 - 5*x^4 + 8*x^3 + 2*x^2 - 4*x + 1)
Time = 0.37 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.22 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {x^2\,{\mathrm {e}}^x\,\left (x^2\,{\mathrm {e}}^x-188\,x-94\,x^2+188\,x^3+94\,x^4+94\right )}{{\left (x^2+x-1\right )}^4} \] Input:
int((exp(x)*(282*x^2 - 188*x + 282*x^3 - 470*x^5 - 376*x^6 + 94*x^7 + 94*x ^8) - exp(2*x)*(4*x^3 + 2*x^4 + 2*x^5 - 2*x^6))/(5*x - 5*x^2 - 10*x^3 + 15 *x^4 + 11*x^5 - 15*x^6 - 10*x^7 + 5*x^8 + 5*x^9 + x^10 - 1),x)
Output:
(x^2*exp(x)*(x^2*exp(x) - 188*x - 94*x^2 + 188*x^3 + 94*x^4 + 94))/(x + x^ 2 - 1)^4
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.11 \[ \int \frac {e^{2 x} \left (-4 x^3-2 x^4-2 x^5+2 x^6\right )+e^x \left (-188 x+282 x^2+282 x^3-470 x^5-376 x^6+94 x^7+94 x^8\right )}{-1+5 x-5 x^2-10 x^3+15 x^4+11 x^5-15 x^6-10 x^7+5 x^8+5 x^9+x^{10}} \, dx=\frac {e^{x} x^{2} \left (e^{x} x^{2}+94 x^{4}+188 x^{3}-94 x^{2}-188 x +94\right )}{x^{8}+4 x^{7}+2 x^{6}-8 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1} \] Input:
int(((2*x^6-2*x^5-2*x^4-4*x^3)*exp(x)^2+(94*x^8+94*x^7-376*x^6-470*x^5+282 *x^3+282*x^2-188*x)*exp(x))/(x^10+5*x^9+5*x^8-10*x^7-15*x^6+11*x^5+15*x^4- 10*x^3-5*x^2+5*x-1),x)
Output:
(e**x*x**2*(e**x*x**2 + 94*x**4 + 188*x**3 - 94*x**2 - 188*x + 94))/(x**8 + 4*x**7 + 2*x**6 - 8*x**5 - 5*x**4 + 8*x**3 + 2*x**2 - 4*x + 1)