Integrand size = 96, antiderivative size = 29 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=\frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (x-(2+x)^2\right )^4} \]
Time = 4.65 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=\frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (4+3 x+x^2\right )^4} \]
Integrate[(40 + 15*x - 7*x^2 + E^(4 + x^2)*(-24 + 12*x^2 - 12*x^3 - 4*x^5 - 6*x^6 - 2*x^7))/(1024 + 3840*x + 7040*x^2 + 8160*x^3 + 6580*x^4 + 3843*x ^5 + 1645*x^6 + 510*x^7 + 110*x^8 + 15*x^9 + x^10),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.
| \(\displaystyle \int \frac {-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40}{x^{10}+15 x^9+110 x^8+510 x^7+1645 x^6+3843 x^5+6580 x^4+8160 x^3+7040 x^2+3840 x+1024} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {20 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \sqrt {7} \left (-2 x+i \sqrt {7}-3\right )}+\frac {20 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \sqrt {7} \left (2 x+i \sqrt {7}+3\right )}-\frac {20 \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \left (-2 x+i \sqrt {7}-3\right )^2}-\frac {20 \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {120 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \sqrt {7} \left (-2 x+i \sqrt {7}-3\right )^3}-\frac {120 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \sqrt {7} \left (2 x+i \sqrt {7}+3\right )^3}+\frac {80 \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \left (-2 x+i \sqrt {7}-3\right )^4}+\frac {80 \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{343 \left (2 x+i \sqrt {7}+3\right )^4}+\frac {32 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{49 \sqrt {7} \left (-2 x+i \sqrt {7}-3\right )^5}+\frac {32 i \left (-7 x^2+e^{x^2+4} \left (-2 x^7-6 x^6-4 x^5-12 x^3+12 x^2-24\right )+15 x+40\right )}{49 \sqrt {7} \left (2 x+i \sqrt {7}+3\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-7 x^2-2 e^{x^2+4} \left (x^7+3 x^6+2 x^5+6 x^3-6 x^2+12\right )+15 x+40}{\left (x^2+3 x+4\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {7 x^2}{\left (x^2+3 x+4\right )^5}+\frac {15 x}{\left (x^2+3 x+4\right )^5}+\frac {40}{\left (x^2+3 x+4\right )^5}-\frac {2 e^{x^2+4} (x+1) \left (x^6+2 x^5+6 x^2-12 x+12\right )}{\left (x^2+3 x+4\right )^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 (2 x+3)}{12 \left (x^2+3 x+4\right )^3}+\frac {10 (2 x+3)}{7 \left (x^2+3 x+4\right )^4}-\frac {10}{49} \left (5 i+\sqrt {7}\right ) \left (i+3 \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {2}{49} \left (3 i+\sqrt {7}\right )^2 e^4 \sqrt {\pi } \text {erfi}(x)+\frac {86}{147} \left (9-5 i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {86}{343} \left (15+i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)-\frac {17}{49} \left (9+i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {20}{49} \left (5+i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)-\frac {107}{147} \left (3+i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {86}{343} \left (15-i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)-\frac {17}{49} \left (9-i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {20}{49} \left (5-i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)-\frac {43}{294} \left (3-i \sqrt {7}\right )^3 e^4 \sqrt {\pi } \text {erfi}(x)-\frac {107}{147} \left (3-i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)-\frac {160}{49} \left (1-i \sqrt {7}\right ) e^4 \sqrt {\pi } \text {erfi}(x)+\frac {2}{49} \left (3 i-\sqrt {7}\right )^2 e^4 \sqrt {\pi } \text {erfi}(x)+\frac {1004}{343} e^4 \sqrt {\pi } \text {erfi}(x)+\frac {43}{49} \left (11 i+9 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {4}{7} \left (9 i+5 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {10}{21} \left (31 i+3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {100}{49} \left (5 i+3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {332}{147} \left (i+3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {125}{49} \left (3 i+\sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {125}{49} \left (3 i-\sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {10}{21} \left (31 i-3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {100}{49} \left (5 i-3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {332}{147} \left (i-3 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {4}{7} \left (9 i-5 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)-\frac {43}{49} \left (11 i-9 \sqrt {7}\right ) e^4 \sqrt {\frac {\pi }{7}} \text {erfi}(x)+\frac {\left (3 i+\sqrt {7}\right )^3 \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx}{\sqrt {7}}+\frac {100}{49} \left (3 i+\sqrt {7}\right )^2 \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {40}{147} \left (77+57 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {16}{49} \left (21-31 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {172}{343} \left (133-15 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {100}{343} \left (7+9 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {344}{343} \left (19-9 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {172}{343} \left (7+6 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {16}{49} \left (9+5 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {344}{147} \left (31+3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {24}{49} \left (7+3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {400}{343} \left (49-3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {800}{343} \left (7-3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {136}{49} \left (5-3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {444}{343} \left (21+i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx-\frac {640}{49} \left (5+i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {948}{343} \left (3-i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {480}{49} \left (1-i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx+\frac {852 i \int \frac {e^{x^2+4}}{-2 x+i \sqrt {7}-3}dx}{49 \sqrt {7}}+\frac {40}{147} \left (77-57 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {16}{49} \left (21+31 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {172}{343} \left (133+15 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {4}{7} \left (35+9 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {344}{343} \left (19+9 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {100}{343} \left (7-9 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {172}{343} \left (7-6 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {16}{49} \left (9-5 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {400}{343} \left (49+3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {800}{343} \left (7+3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {136}{49} \left (5+3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {344}{147} \left (31-3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {24}{49} \left (7-3 i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {948}{343} \left (3+i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {480}{49} \left (1+i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {444}{343} \left (21-i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {640}{49} \left (5-i \sqrt {7}\right ) \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx-\frac {100}{49} \left (3 i-\sqrt {7}\right )^2 \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx+\frac {852 i \int \frac {e^{x^2+4}}{2 x+i \sqrt {7}+3}dx}{49 \sqrt {7}}-\frac {4 \left (3 i+\sqrt {7}\right )^2 e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )}-\frac {20 \left (21-31 i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )}-\frac {86 \left (63+11 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}+\frac {8 \left (35-9 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )}+\frac {200 \left (21+5 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}-\frac {250 \left (7+3 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}+\frac {52 \left (21+i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )}+\frac {320 \left (1+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )}+\frac {100 \left (21-i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}-\frac {172 \left (15-i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}+\frac {34 \left (9-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )}-\frac {40 \left (5-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )}+\frac {43 \left (3-i \sqrt {7}\right )^3 e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )}+\frac {214 \left (3-i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )}-\frac {1004 e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )}-\frac {20 \left (21+31 i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )}-\frac {86 \left (63-11 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}+\frac {8 \left (35+9 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )}+\frac {200 \left (21-5 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}-\frac {172 \left (9-5 i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )}-\frac {250 \left (7-3 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}+\frac {100 \left (21+i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}-\frac {172 \left (15+i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}+\frac {34 \left (9+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )}-\frac {40 \left (5+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )}+\frac {214 \left (3+i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )}+\frac {52 \left (21-i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )}+\frac {320 \left (1-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )}-\frac {4 \left (3 i-\sqrt {7}\right )^2 e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )}-\frac {1004 e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )}-\frac {5 \left (3 i+\sqrt {7}\right )^3 e^{x^2+4}}{21 \sqrt {7} \left (2 x-i \sqrt {7}+3\right )^2}+\frac {86 \left (3 i+\sqrt {7}\right )^2 e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )^2}-\frac {100 \left (7+9 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )^2}+\frac {172 \left (7+6 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x-i \sqrt {7}+3\right )^2}-\frac {24 \left (7+3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^2}-\frac {8 \left (21+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^2}-\frac {8 \left (3-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^2}+\frac {160 \left (1-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^2}+\frac {612 i e^{x^2+4}}{49 \sqrt {7} \left (2 x-i \sqrt {7}+3\right )^2}+\frac {20 \left (35+9 i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {100 \left (7-9 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )^2}+\frac {172 \left (7-6 i \sqrt {7}\right ) e^{x^2+4}}{343 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {24 \left (7-3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {8 \left (3+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^2}+\frac {160 \left (1+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {8 \left (21-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^2}+\frac {86 \left (3 i-\sqrt {7}\right )^2 e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )^2}-\frac {612 i e^{x^2+4}}{49 \sqrt {7} \left (2 x+i \sqrt {7}+3\right )^2}+\frac {16 \left (7+3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^3}-\frac {40 \left (21+i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )^3}-\frac {80 \left (5-i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^3}+\frac {344 \left (3-i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x-i \sqrt {7}+3\right )^3}+\frac {32 e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^3}+\frac {16 \left (7-3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^3}-\frac {80 \left (5+i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^3}+\frac {344 \left (3+i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )^3}-\frac {40 \left (21-i \sqrt {7}\right ) e^{x^2+4}}{147 \left (2 x+i \sqrt {7}+3\right )^3}+\frac {32 e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^3}-\frac {5 x+12}{6 \left (x^2+3 x+4\right )^3}+\frac {40 \left (7+3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x-i \sqrt {7}+3\right )^4}-\frac {96 i e^{x^2+4}}{7 \sqrt {7} \left (2 x-i \sqrt {7}+3\right )^4}+\frac {40 \left (7-3 i \sqrt {7}\right ) e^{x^2+4}}{49 \left (2 x+i \sqrt {7}+3\right )^4}+\frac {96 i e^{x^2+4}}{7 \sqrt {7} \left (2 x+i \sqrt {7}+3\right )^4}+\frac {x (3 x+8)}{4 \left (x^2+3 x+4\right )^4}-\frac {15 (3 x+8)}{28 \left (x^2+3 x+4\right )^4}\) |
Int[(40 + 15*x - 7*x^2 + E^(4 + x^2)*(-24 + 12*x^2 - 12*x^3 - 4*x^5 - 6*x^ 6 - 2*x^7))/(1024 + 3840*x + 7040*x^2 + 8160*x^3 + 6580*x^4 + 3843*x^5 + 1 645*x^6 + 510*x^7 + 110*x^8 + 15*x^9 + x^10),x]
3.1.15.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17
| method | result | size |
| norman | \(\frac {x -3-{\mathrm e}^{x^{2}+4} x^{4}+2 \,{\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) | \(34\) |
| parts | \(-\frac {-x +3}{\left (x^{2}+3 x +4\right )^{4}}+\frac {-{\mathrm e}^{x^{2}+4} x^{4}+2 \,{\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) | \(50\) |
| parallelrisch | \(\frac {x -3-{\mathrm e}^{x^{2}+4} x^{4}+2 \,{\mathrm e}^{x^{2}+4}}{x^{8}+12 x^{7}+70 x^{6}+252 x^{5}+609 x^{4}+1008 x^{3}+1120 x^{2}+768 x +256}\) | \(64\) |
| risch | \(\frac {-3+x}{x^{8}+12 x^{7}+70 x^{6}+252 x^{5}+609 x^{4}+1008 x^{3}+1120 x^{2}+768 x +256}-\frac {\left (x^{4}-2\right ) {\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) | \(69\) |
int(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40)/(x^10 +15*x^9+110*x^8+510*x^7+1645*x^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2+3840* x+1024),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=-\frac {{\left (x^{4} - 2\right )} e^{\left (x^{2} + 4\right )} - x + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40) /(x^10+15*x^9+110*x^8+510*x^7+1645*x^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2 +3840*x+1024),x, algorithm=\
-((x^4 - 2)*e^(x^2 + 4) - x + 3)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^ 4 + 1008*x^3 + 1120*x^2 + 768*x + 256)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=\frac {\left (2 - x^{4}\right ) e^{x^{2} + 4}}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} - \frac {3 - x}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} \]
integrate(((-2*x**7-6*x**6-4*x**5-12*x**3+12*x**2-24)*exp(x**2+4)-7*x**2+1 5*x+40)/(x**10+15*x**9+110*x**8+510*x**7+1645*x**6+3843*x**5+6580*x**4+816 0*x**3+7040*x**2+3840*x+1024),x)
(2 - x**4)*exp(x**2 + 4)/(x**8 + 12*x**7 + 70*x**6 + 252*x**5 + 609*x**4 + 1008*x**3 + 1120*x**2 + 768*x + 256) - (3 - x)/(x**8 + 12*x**7 + 70*x**6 + 252*x**5 + 609*x**4 + 1008*x**3 + 1120*x**2 + 768*x + 256)
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (30) = 60\).
Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 9.97 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=-\frac {{\left (x^{4} e^{4} - 2 \, e^{4}\right )} e^{\left (x^{2}\right )}}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} - \frac {300 \, x^{7} + 3150 \, x^{6} + 16100 \, x^{5} + 49875 \, x^{4} + 100940 \, x^{3} + 132930 \, x^{2} + 106512 \, x + 41904}{588 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} + \frac {10 \, {\left (120 \, x^{7} + 1260 \, x^{6} + 6440 \, x^{5} + 19950 \, x^{4} + 40376 \, x^{3} + 53172 \, x^{2} + 42840 \, x + 16497\right )}}{1029 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} - \frac {15 \, {\left (60 \, x^{7} + 630 \, x^{6} + 3220 \, x^{5} + 9975 \, x^{4} + 20188 \, x^{3} + 26586 \, x^{2} + 21420 \, x + 8420\right )}}{1372 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} \]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40) /(x^10+15*x^9+110*x^8+510*x^7+1645*x^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2 +3840*x+1024),x, algorithm=\
-(x^4*e^4 - 2*e^4)*e^(x^2)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 10 08*x^3 + 1120*x^2 + 768*x + 256) - 1/588*(300*x^7 + 3150*x^6 + 16100*x^5 + 49875*x^4 + 100940*x^3 + 132930*x^2 + 106512*x + 41904)/(x^8 + 12*x^7 + 7 0*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256) + 10/1029*( 120*x^7 + 1260*x^6 + 6440*x^5 + 19950*x^4 + 40376*x^3 + 53172*x^2 + 42840* x + 16497)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^ 2 + 768*x + 256) - 15/1372*(60*x^7 + 630*x^6 + 3220*x^5 + 9975*x^4 + 20188 *x^3 + 26586*x^2 + 21420*x + 8420)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609* x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=-\frac {x^{4} e^{\left (x^{2} + 4\right )} - x - 2 \, e^{\left (x^{2} + 4\right )} + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40) /(x^10+15*x^9+110*x^8+510*x^7+1645*x^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2 +3840*x+1024),x, algorithm=\
-(x^4*e^(x^2 + 4) - x - 2*e^(x^2 + 4) + 3)/(x^8 + 12*x^7 + 70*x^6 + 252*x^ 5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256)
Time = 0.40 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx=\frac {x+2\,{\mathrm {e}}^{x^2+4}-x^4\,{\mathrm {e}}^{x^2+4}-3}{{\left (x^2+3\,x+4\right )}^4} \]