Integrand size = 120, antiderivative size = 24 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=e^x+x^2-\frac {75}{x \left (-7-x-x^2\right )^2} \]
Time = 3.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=e^x+\frac {-75+49 x^3+14 x^4+15 x^5+2 x^6+x^7}{x \left (7+x+x^2\right )^2} \]
Integrate[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^ 6 + 3*x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8),x]
Result contains complex when optimal does not.
Time = 11.80 (sec) , antiderivative size = 6015, normalized size of antiderivative = 250.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2026, 2463, 2009}
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.
| \(\displaystyle \int \frac {2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525}{x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525}{x^2 \left (x^6+3 x^5+24 x^4+43 x^3+168 x^2+147 x+343\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{729 \sqrt {3} x^2 \left (2 x+3 i \sqrt {3}+1\right )}+\frac {4 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{729 \sqrt {3} \left (-2 x+3 i \sqrt {3}-1\right ) x^2}-\frac {4 \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{243 \left (-2 x+3 i \sqrt {3}-1\right )^2 x^2}-\frac {8 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{81 \sqrt {3} \left (-2 x+3 i \sqrt {3}-1\right )^3 x^2}-\frac {4 \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{243 x^2 \left (2 x+3 i \sqrt {3}+1\right )^2}-\frac {8 i \left (2 x^9+6 x^8+48 x^7+86 x^6+336 x^5+294 x^4+686 x^3+375 x^2+e^x \left (x^8+3 x^7+24 x^6+43 x^5+168 x^4+147 x^3+343 x^2\right )+225 x+525\right )}{81 \sqrt {3} x^2 \left (2 x+3 i \sqrt {3}+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (9+i \sqrt {3}\right ) x^6}{6561}+\frac {\left (9-i \sqrt {3}\right ) x^6}{6561}-\frac {2 x^6}{729}-\frac {2 \left (9+13 i \sqrt {3}\right ) x^5}{10935}-\frac {2 \left (9-13 i \sqrt {3}\right ) x^5}{10935}+\frac {2 \left (1+3 i \sqrt {3}\right ) x^5}{1215}+\frac {2 \left (1-3 i \sqrt {3}\right ) x^5}{1215}+\frac {2 \left (9+i \sqrt {3}\right ) x^5}{3645}+\frac {2 \left (9-i \sqrt {3}\right ) x^5}{3645}-\frac {4 x^5}{405}-\frac {\left (9+13 i \sqrt {3}\right ) x^4}{1458}-\frac {\left (9-13 i \sqrt {3}\right ) x^4}{1458}-\frac {\left (27+10 i \sqrt {3}\right ) x^4}{2187}-\frac {\left (27-10 i \sqrt {3}\right ) x^4}{2187}+\frac {1}{324} \left (13+3 i \sqrt {3}\right ) x^4+\frac {1}{162} \left (1+3 i \sqrt {3}\right ) x^4+\frac {1}{324} \left (13-3 i \sqrt {3}\right ) x^4+\frac {1}{162} \left (1-3 i \sqrt {3}\right ) x^4+\frac {7 \left (9+i \sqrt {3}\right ) x^4}{2916}+\frac {7 \left (9-i \sqrt {3}\right ) x^4}{2916}-\frac {8 x^4}{81}+\frac {2 \left (117+71 i \sqrt {3}\right ) x^3}{6561}+\frac {2 \left (117-71 i \sqrt {3}\right ) x^3}{6561}+\frac {2 \left (9+13 i \sqrt {3}\right ) x^3}{2187}+\frac {2 \left (9-13 i \sqrt {3}\right ) x^3}{2187}-\frac {4 \left (27+10 i \sqrt {3}\right ) x^3}{2187}-\frac {4 \left (27-10 i \sqrt {3}\right ) x^3}{2187}-\frac {8}{729} \left (10+9 i \sqrt {3}\right ) x^3-\frac {8}{729} \left (10-9 i \sqrt {3}\right ) x^3+\frac {1}{81} \left (13+3 i \sqrt {3}\right ) x^3+\frac {16}{243} \left (1+3 i \sqrt {3}\right ) x^3+\frac {1}{81} \left (13-3 i \sqrt {3}\right ) x^3+\frac {16}{243} \left (1-3 i \sqrt {3}\right ) x^3+\frac {5 \left (9+i \sqrt {3}\right ) x^3}{6561}+\frac {5 \left (9-i \sqrt {3}\right ) x^3}{6561}-\frac {172 x^3}{729}-\frac {2744 x^2}{81 \sqrt {3} \left (i-3 \sqrt {3}\right ) \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {2744 x^2}{81 \sqrt {3} \left (i+3 \sqrt {3}\right ) \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {\left (i+3 \sqrt {3}\right )^5 x^2}{11664 \sqrt {3}}-\frac {5 \left (i+3 \sqrt {3}\right )^4 x^2}{3888}+\frac {1}{729} \left (117+71 i \sqrt {3}\right ) x^2+\frac {1}{729} \left (117-71 i \sqrt {3}\right ) x^2-\frac {5}{486} \left (71-39 i \sqrt {3}\right ) x^2+\frac {38 \left (9+13 i \sqrt {3}\right ) x^2}{2187}+\frac {38 \left (9-13 i \sqrt {3}\right ) x^2}{2187}+\frac {14}{729} \left (27+10 i \sqrt {3}\right ) x^2+\frac {14}{729} \left (27-10 i \sqrt {3}\right ) x^2-\frac {4}{81} \left (10+9 i \sqrt {3}\right ) x^2-\frac {4}{81} \left (10-9 i \sqrt {3}\right ) x^2+\frac {4}{27} \left (13+3 i \sqrt {3}\right ) x^2+\frac {43}{243} \left (1+3 i \sqrt {3}\right ) x^2+\frac {4}{27} \left (13-3 i \sqrt {3}\right ) x^2+\frac {43}{243} \left (1-3 i \sqrt {3}\right ) x^2-\frac {52}{729} \left (9+i \sqrt {3}\right ) x^2-\frac {52}{729} \left (9-i \sqrt {3}\right ) x^2-\frac {\left (i-3 \sqrt {3}\right )^5 x^2}{11664 \sqrt {3}}-\frac {112 x^2}{81}+\frac {\left (i+3 \sqrt {3}\right )^5 x}{1944 \sqrt {3}}-\frac {5}{648} \left (i+3 \sqrt {3}\right )^4 x+\frac {2}{729} \left (261-211 i \sqrt {3}\right ) x-\frac {4 \left (540+143 i \sqrt {3}\right ) x}{2187}-\frac {4 \left (540-143 i \sqrt {3}\right ) x}{2187}-\frac {29}{729} \left (117+71 i \sqrt {3}\right ) x-\frac {29}{729} \left (117-71 i \sqrt {3}\right ) x-\frac {5}{81} \left (71-39 i \sqrt {3}\right ) x+\frac {320}{729} \left (9+13 i \sqrt {3}\right ) x+\frac {320}{729} \left (9-13 i \sqrt {3}\right ) x+\frac {368 \left (27+10 i \sqrt {3}\right ) x}{2187}+\frac {368 \left (27-10 i \sqrt {3}\right ) x}{2187}-\frac {64}{81} \left (10+9 i \sqrt {3}\right ) x-\frac {64}{81} \left (10-9 i \sqrt {3}\right ) x+\frac {43}{81} \left (13+3 i \sqrt {3}\right ) x+\frac {1}{648} \left (1+3 i \sqrt {3}\right )^5 x+\frac {112}{81} \left (1+3 i \sqrt {3}\right ) x+\frac {43}{81} \left (13-3 i \sqrt {3}\right ) x+\frac {1}{648} \left (1-3 i \sqrt {3}\right )^5 x+\frac {112}{81} \left (1-3 i \sqrt {3}\right ) x-\frac {289}{729} \left (9+i \sqrt {3}\right ) x-\frac {289}{729} \left (9-i \sqrt {3}\right ) x-\frac {196 x}{81}-\frac {80 e^x}{81}-\frac {2 e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^5}{729 \sqrt {3}}-\frac {2 i e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^5}{729 \sqrt {3}}-\frac {10 i e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^4}{729 \sqrt {3}}-\frac {5}{243} e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^4-\frac {1}{243} e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )^4+\frac {10 i e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^4}{729 \sqrt {3}}-\frac {4}{243} e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^4-\frac {41 e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^3}{729 \sqrt {3}}-\frac {20}{243} i e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^3+\frac {4 e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )^3}{81 \sqrt {3}}-\frac {4}{243} i e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )^3+\frac {14 i e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^3}{729 \sqrt {3}}+\frac {16}{243} e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^3-\frac {i e^x \left (x+\frac {14}{1+3 i \sqrt {3}}\right )^3}{81 \sqrt {3}}-\frac {41 i e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^2}{243 \sqrt {3}}+\frac {20}{81} e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )^2+\frac {4 i e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )^2}{27 \sqrt {3}}+\frac {4}{81} e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )^2-\frac {14 i e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^2}{243 \sqrt {3}}-\frac {16}{81} e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )^2+\frac {i e^x \left (x+\frac {14}{1+3 i \sqrt {3}}\right )^2}{27 \sqrt {3}}+\frac {82 e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )}{243 \sqrt {3}}+\frac {40}{81} i e^x \left (\frac {14}{i-3 \sqrt {3}}-i x\right )-\frac {8 e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )}{27 \sqrt {3}}+\frac {8}{81} i e^x \left (\frac {14}{i+3 \sqrt {3}}-i x\right )+\frac {28 i e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )}{243 \sqrt {3}}+\frac {32}{81} e^x \left (x+\frac {14}{1-3 i \sqrt {3}}\right )-\frac {2 i e^x \left (x+\frac {14}{1+3 i \sqrt {3}}\right )}{27 \sqrt {3}}-\frac {2 \left (747-1763 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )}{2187}-\frac {47}{729} \left (261-211 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {14}{243} \left (143-180 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )-\frac {4}{729} \left (540+143 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {2}{27} \left (211+87 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )-\frac {319 \left (117+71 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )}{2187}-\frac {40}{81} \left (71-39 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {676}{729} \left (9+13 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {1216}{729} \left (27-10 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )-\frac {344}{243} \left (10+9 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {98}{81} \left (1+3 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )+\frac {56}{27} \left (13-3 i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )-\frac {3850 \left (9-i \sqrt {3}\right ) \log \left (2 i x-3 \sqrt {3}+i\right )}{2187}-\frac {2800 i \log \left (2 i x-3 \sqrt {3}+i\right )}{9 \sqrt {3} \left (i-3 \sqrt {3}\right )^4}+\frac {25 \log \left (2 i x-3 \sqrt {3}+i\right )}{9 \sqrt {3} \left (10 i-9 \sqrt {3}\right )}-\frac {632 i \log \left (2 i x-3 \sqrt {3}+i\right )}{243 \sqrt {3}}-\frac {686}{243} \log \left (2 i x-3 \sqrt {3}+i\right )-\frac {100 \log \left (2 i x+3 \sqrt {3}+i\right )}{81 \sqrt {3} \left (i+3 \sqrt {3}\right )}-\frac {200 \log \left (2 i x+3 \sqrt {3}+i\right )}{9 \sqrt {3} \left (i+3 \sqrt {3}\right )^3}+\frac {2800 i \log \left (2 i x+3 \sqrt {3}+i\right )}{9 \sqrt {3} \left (i+3 \sqrt {3}\right )^4}-\frac {2 \left (747+1763 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )}{2187}-\frac {47}{729} \left (261+211 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {14}{243} \left (143+180 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )-\frac {4}{729} \left (540-143 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {2}{27} \left (211-87 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )-\frac {319 \left (117-71 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )}{2187}-\frac {40}{81} \left (71+39 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {676}{729} \left (9-13 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {1216}{729} \left (27+10 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )-\frac {344}{243} \left (10-9 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {56}{27} \left (13+3 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {98}{81} \left (1-3 i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )-\frac {3850 \left (9+i \sqrt {3}\right ) \log \left (2 i x+3 \sqrt {3}+i\right )}{2187}+\frac {632 i \log \left (2 i x+3 \sqrt {3}+i\right )}{243 \sqrt {3}}-\frac {686}{243} \log \left (2 i x+3 \sqrt {3}+i\right )+\frac {25 \log (x)}{9 \sqrt {3} \left (10 i+9 \sqrt {3}\right )}-\frac {700 \log (x)}{243 \sqrt {3} \left (13 i+3 \sqrt {3}\right )}+\frac {100 \log (x)}{81 \sqrt {3} \left (i+3 \sqrt {3}\right )}+\frac {100 \log (x)}{27 \left (i+3 \sqrt {3}\right )^2}-\frac {2800 i \log (x)}{9 \sqrt {3} \left (i+3 \sqrt {3}\right )^4}-\frac {350 \log (x)}{81 \left (10+9 i \sqrt {3}\right )}+\frac {2800 \log (x)}{81 \left (1-3 i \sqrt {3}\right )^3}+\frac {700 \log (x)}{243 \sqrt {3} \left (13 i-3 \sqrt {3}\right )}-\frac {100 \log (x)}{81 \sqrt {3} \left (i-3 \sqrt {3}\right )}+\frac {100 \log (x)}{27 \left (i-3 \sqrt {3}\right )^2}+\frac {2800 i \log (x)}{9 \sqrt {3} \left (i-3 \sqrt {3}\right )^4}-\frac {25 \log (x)}{9 \sqrt {3} \left (10 i-9 \sqrt {3}\right )}+\frac {100 \log \left (2 x+3 i \sqrt {3}+1\right )}{81 \sqrt {3} \left (i-3 \sqrt {3}\right )}+\frac {350 \log \left (-2 \left (10 i-9 \sqrt {3}\right ) x+39 \sqrt {3}+71 i\right )}{81 \left (10+9 i \sqrt {3}\right )}-\frac {50 \log \left (\left (13 i-3 \sqrt {3}\right ) x+2 \left (10 i+9 \sqrt {3}\right )\right )}{27 \left (13+3 i \sqrt {3}\right )}-\frac {700 \log \left (\left (13 i-3 \sqrt {3}\right ) x+2 \left (10 i+9 \sqrt {3}\right )\right )}{243 \sqrt {3} \left (13 i-3 \sqrt {3}\right )}+\frac {700 \log \left (-\left (\left (13 i+3 \sqrt {3}\right ) x\right )-2 \left (10 i-9 \sqrt {3}\right )\right )}{243 \sqrt {3} \left (13 i+3 \sqrt {3}\right )}-\frac {50 \log \left (-\left (\left (13 i+3 \sqrt {3}\right ) x\right )-2 \left (10 i-9 \sqrt {3}\right )\right )}{27 \left (13-3 i \sqrt {3}\right )}+\frac {350 \log \left (-2 \left (10 i+9 \sqrt {3}\right ) x-39 \sqrt {3}+71 i\right )}{81 \left (10-9 i \sqrt {3}\right )}+\frac {4 \left (261 i+211 \sqrt {3}\right )}{27 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {4 \left (143 i+180 \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {86 \left (71 i+39 \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {688 \left (27 i+10 \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {98 \left (13 i+3 \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {196 \left (9 i+\sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {5600}{27 \sqrt {3} \left (1+3 i \sqrt {3}\right )^3 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {686 \left (i-3 \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {100}{27 \left (i-3 \sqrt {3}\right ) \left (2 i x-3 \sqrt {3}+i\right )}+\frac {224 \left (10 i-9 \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {112 \left (9 i-13 \sqrt {3}\right )}{27 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {80 \left (117 i-71 \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {16 \left (211 i-87 \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )}-\frac {28 \left (540 i-143 \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {2 \left (1763 i-249 \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {250 i}{81 \left (2 i x-3 \sqrt {3}+i\right )}+\frac {2 \left (1763 i+249 \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {28 \left (540 i+143 \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {16 \left (211 i+87 \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {80 \left (117 i+71 \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {112 \left (9 i+13 \sqrt {3}\right )}{27 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {224 \left (10 i+9 \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {686 \left (i+3 \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {100}{27 \left (i+3 \sqrt {3}\right ) \left (2 i x+3 \sqrt {3}+i\right )}+\frac {5600}{27 \sqrt {3} \left (1-3 i \sqrt {3}\right )^3 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {196 \left (9 i-\sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {98 \left (13 i-3 \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {688 \left (27 i-10 \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {86 \left (71 i-39 \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {4 \left (143 i-180 \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {4 \left (261 i-211 \sqrt {3}\right )}{27 \left (2 i x+3 \sqrt {3}+i\right )}+\frac {250 i}{81 \left (2 i x+3 \sqrt {3}+i\right )}-\frac {100}{9 \sqrt {3} \left (1+3 i \sqrt {3}\right ) \left (-\left (\left (i-3 \sqrt {3}\right ) x\right )+3 \sqrt {3}+13 i\right )}+\frac {700}{81 \left (i-3 \sqrt {3}\right ) \left (-\left (\left (i-3 \sqrt {3}\right ) x\right )+3 \sqrt {3}+13 i\right )}+\frac {700}{81 \left (i+3 \sqrt {3}\right ) \left (-\left (\left (i+3 \sqrt {3}\right ) x\right )-3 \sqrt {3}+13 i\right )}+\frac {100}{9 \sqrt {3} \left (1-3 i \sqrt {3}\right ) \left (-\left (\left (i+3 \sqrt {3}\right ) x\right )-3 \sqrt {3}+13 i\right )}-\frac {700}{27 \sqrt {3} \left (13 i+3 \sqrt {3}\right ) \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {2 \left (747-1763 i \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )^2}-\frac {16 \left (261-211 i \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {4 \left (540+143 i \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )^2}-\frac {86 \left (117+71 i \sqrt {3}\right )}{243 \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {98 \left (9+13 i \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {224 \left (27-10 i \sqrt {3}\right )}{81 \left (2 i x-3 \sqrt {3}+i\right )^2}-\frac {100}{9 \sqrt {3} \left (i-3 \sqrt {3}\right ) \left (2 i x-3 \sqrt {3}+i\right )^2}-\frac {250 i}{27 \sqrt {3} \left (2 i x-3 \sqrt {3}+i\right )^2}+\frac {100}{9 \sqrt {3} \left (i+3 \sqrt {3}\right ) \left (2 i x+3 \sqrt {3}+i\right )^2}-\frac {1400 i}{27 \sqrt {3} \left (i+3 \sqrt {3}\right )^2 \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {2 \left (747+1763 i \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )^2}-\frac {16 \left (261+211 i \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {4 \left (540-143 i \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )^2}-\frac {86 \left (117-71 i \sqrt {3}\right )}{243 \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {98 \left (9-13 i \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {224 \left (27+10 i \sqrt {3}\right )}{81 \left (2 i x+3 \sqrt {3}+i\right )^2}+\frac {250 i}{27 \sqrt {3} \left (2 i x+3 \sqrt {3}+i\right )^2}-\frac {700}{243 \sqrt {3} \left (i+3 \sqrt {3}\right ) x}-\frac {700}{81 \left (i+3 \sqrt {3}\right )^2 x}-\frac {1400}{27 \sqrt {3} \left (i+3 \sqrt {3}\right )^3 x}+\frac {700}{243 \sqrt {3} \left (i-3 \sqrt {3}\right ) x}-\frac {700}{81 \left (i-3 \sqrt {3}\right )^2 x}+\frac {175}{27 \sqrt {3} \left (10 i-9 \sqrt {3}\right ) x}\) |
Int[(525 + 225*x + 375*x^2 + 686*x^3 + 294*x^4 + 336*x^5 + 86*x^6 + 48*x^7 + 6*x^8 + 2*x^9 + E^x*(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3* x^7 + x^8))/(343*x^2 + 147*x^3 + 168*x^4 + 43*x^5 + 24*x^6 + 3*x^7 + x^8), x]
(-80*E^x)/81 + ((40*I)/81)*E^x*(14/(I - 3*Sqrt[3]) - I*x) + (82*E^x*(14/(I - 3*Sqrt[3]) - I*x))/(243*Sqrt[3]) + (20*E^x*(14/(I - 3*Sqrt[3]) - I*x)^2 )/81 - (((41*I)/243)*E^x*(14/(I - 3*Sqrt[3]) - I*x)^2)/Sqrt[3] - ((20*I)/2 43)*E^x*(14/(I - 3*Sqrt[3]) - I*x)^3 - (41*E^x*(14/(I - 3*Sqrt[3]) - I*x)^ 3)/(729*Sqrt[3]) - (5*E^x*(14/(I - 3*Sqrt[3]) - I*x)^4)/243 - (((10*I)/729 )*E^x*(14/(I - 3*Sqrt[3]) - I*x)^4)/Sqrt[3] - (2*E^x*(14/(I - 3*Sqrt[3]) - I*x)^5)/(729*Sqrt[3]) + ((8*I)/81)*E^x*(14/(I + 3*Sqrt[3]) - I*x) - (8*E^ x*(14/(I + 3*Sqrt[3]) - I*x))/(27*Sqrt[3]) + (4*E^x*(14/(I + 3*Sqrt[3]) - I*x)^2)/81 + (((4*I)/27)*E^x*(14/(I + 3*Sqrt[3]) - I*x)^2)/Sqrt[3] - ((4*I )/243)*E^x*(14/(I + 3*Sqrt[3]) - I*x)^3 + (4*E^x*(14/(I + 3*Sqrt[3]) - I*x )^3)/(81*Sqrt[3]) - (E^x*(14/(I + 3*Sqrt[3]) - I*x)^4)/243 - ((250*I)/27)/ (Sqrt[3]*(I - 3*Sqrt[3] + (2*I)*x)^2) - 100/(9*Sqrt[3]*(I - 3*Sqrt[3])*(I - 3*Sqrt[3] + (2*I)*x)^2) + (224*(27 - (10*I)*Sqrt[3]))/(81*(I - 3*Sqrt[3] + (2*I)*x)^2) + (98*(9 + (13*I)*Sqrt[3]))/(81*(I - 3*Sqrt[3] + (2*I)*x)^2 ) - (86*(117 + (71*I)*Sqrt[3]))/(243*(I - 3*Sqrt[3] + (2*I)*x)^2) + (4*(54 0 + (143*I)*Sqrt[3]))/(81*(I - 3*Sqrt[3] + (2*I)*x)^2) - (16*(261 - (211*I )*Sqrt[3]))/(81*(I - 3*Sqrt[3] + (2*I)*x)^2) + (2*(747 - (1763*I)*Sqrt[3]) )/(243*(I - 3*Sqrt[3] + (2*I)*x)^2) - 700/(27*Sqrt[3]*(13*I + 3*Sqrt[3])*( I - 3*Sqrt[3] + (2*I)*x)^2) + ((250*I)/81)/(I - 3*Sqrt[3] + (2*I)*x) + (2* (1763*I - 249*Sqrt[3]))/(243*(I - 3*Sqrt[3] + (2*I)*x)) - (28*(540*I - ...
3.11.75.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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]
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(x^{2}-\frac {75}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+{\mathrm e}^{x}\) | \(32\) |
| parts | \(x^{2}-\frac {75}{49 x}-\frac {75 \left (-x^{3}-2 x^{2}-15 x -14\right )}{49 \left (x^{2}+x +7\right )^{2}}+{\mathrm e}^{x}\) | \(37\) |
| norman | \(\frac {-75+x^{7}+x^{5} {\mathrm e}^{x}-735 x -210 x^{2}-176 x^{3}-16 x^{4}+2 x^{6}+49 \,{\mathrm e}^{x} x +14 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{x} x^{4}}{x \left (x^{2}+x +7\right )^{2}}\) | \(73\) |
| parallelrisch | \(\frac {-75+x^{7}+x^{5} {\mathrm e}^{x}-735 x -210 x^{2}-176 x^{3}-16 x^{4}+2 x^{6}+49 \,{\mathrm e}^{x} x +14 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{x} x^{4}}{x \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}\) | \(85\) |
| default | \(\frac {\frac {125}{18}+\frac {125 x}{9}}{\left (x^{2}+x +7\right )^{2}}+\frac {\frac {32}{243}+\frac {64 x}{243}}{x^{2}+x +7}-\frac {75}{49 x}+x^{2}+{\mathrm e}^{x}-\frac {4 \,{\mathrm e}^{x} \left (173 x^{3}-546 x^{2}-147 x -3773\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}-\frac {43 \,{\mathrm e}^{x} \left (68 x^{3}+255 x^{2}+483 x +490\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}-\frac {225 \left (\frac {112}{243} x^{3}-\frac {455}{162} x^{2}+\frac {70}{81} x -\frac {8428}{243}\right )}{343 \left (x^{2}+x +7\right )^{2}}+\frac {28 \,{\mathrm e}^{x} \left (x^{3}-57 x^{2}-168 x -343\right )}{81 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {-\frac {343 x}{27}-\frac {4802}{27}}{\left (x^{2}+x +7\right )^{2}}+\frac {49 \,{\mathrm e}^{x} \left (7 x^{3}+6 x^{2}+39 x -133\right )}{162 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {\frac {490}{81} x^{3}+\frac {245}{27} x^{2}-\frac {686}{27} x +\frac {4802}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {12470}{243} x^{3}+\frac {731}{81} x^{2}-\frac {15050}{81} x +\frac {48461}{243}}{\left (x^{2}+x +7\right )^{2}}+\frac {343 \,{\mathrm e}^{x} \left (4 x^{3}+15 x^{2}+57 x +86\right )}{486 \left (x^{4}+2 x^{3}+15 x^{2}+14 x +49\right )}+\frac {\frac {5984}{81} x^{3}+\frac {10120}{27} x^{2}+\frac {17024}{27} x +\frac {134848}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {{\mathrm e}^{x} \left (1009 x^{3}+3237 x^{2}+7392 x +11123\right )}{162 x^{4}+324 x^{3}+2430 x^{2}+2268 x +7938}-\frac {2 \left (\frac {5797}{243} x^{3}+\frac {11305}{162} x^{2}+\frac {14602}{81} x +\frac {138229}{486}\right )}{\left (x^{2}+x +7\right )^{2}}+\frac {{\mathrm e}^{x} \left (1480 x^{3}-8301 x^{2}-1995 x -55174\right )}{486 x^{4}+972 x^{3}+7290 x^{2}+6804 x +23814}-\frac {6 \left (-\frac {1280}{243} x^{3}+\frac {899}{81} x^{2}-\frac {1232}{81} x +\frac {42385}{486}\right )}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {784}{81} x^{3}-\frac {4928}{27} x^{2}-\frac {3136}{27} x -\frac {52136}{81}}{\left (x^{2}+x +7\right )^{2}}+\frac {-\frac {4975}{3969} x^{3}-\frac {9025}{2646} x^{2}-\frac {17875}{1323} x -\frac {21625}{1134}}{\left (x^{2}+x +7\right )^{2}}\) | \(528\) |
int(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+6*x^8+ 48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+24*x^6 +43*x^5+168*x^4+147*x^3+343*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=\frac {x^{7} + 2 \, x^{6} + 15 \, x^{5} + 14 \, x^{4} + 49 \, x^{3} + {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )} e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \]
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm=\
(x^7 + 2*x^6 + 15*x^5 + 14*x^4 + 49*x^3 + (x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)*e^x - 75)/(x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=x^{2} + e^{x} - \frac {75}{x^{5} + 2 x^{4} + 15 x^{3} + 14 x^{2} + 49 x} \]
integrate(((x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2)*exp(x) +2*x**9+6*x**8+48*x**7+86*x**6+336*x**5+294*x**4+686*x**3+375*x**2+225*x+5 25)/(x**8+3*x**7+24*x**6+43*x**5+168*x**4+147*x**3+343*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (19) = 38\).
Time = 0.31 (sec) , antiderivative size = 385, normalized size of antiderivative = 16.04 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=x^{2} - \frac {25 \, {\left (884 \, x^{4} + 2055 \, x^{3} + 11580 \, x^{2} + 12859 \, x + 23814\right )}}{7938 \, {\left (x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x\right )}} - \frac {11594 \, x^{3} + 33915 \, x^{2} + 87612 \, x + 138229}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {2560 \, x^{3} - 5394 \, x^{2} + 7392 \, x - 42385}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {8 \, {\left (748 \, x^{3} + 3795 \, x^{2} + 6384 \, x + 16856\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {43 \, {\left (290 \, x^{3} - 51 \, x^{2} + 1050 \, x - 1127\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {25 \, {\left (32 \, x^{3} - 195 \, x^{2} + 60 \, x - 2408\right )}}{2646 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {56 \, {\left (14 \, x^{3} + 264 \, x^{2} + 168 \, x + 931\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {49 \, {\left (10 \, x^{3} + 15 \, x^{2} - 42 \, x + 98\right )}}{81 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + \frac {125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 48 \, x + 23\right )}}{162 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} - \frac {343 \, {\left (2 \, x^{3} + 3 \, x^{2} + 24 \, x + 133\right )}}{243 \, {\left (x^{4} + 2 \, x^{3} + 15 \, x^{2} + 14 \, x + 49\right )}} + e^{x} \]
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm=\
x^2 - 25/7938*(884*x^4 + 2055*x^3 + 11580*x^2 + 12859*x + 23814)/(x^5 + 2* x^4 + 15*x^3 + 14*x^2 + 49*x) - 1/243*(11594*x^3 + 33915*x^2 + 87612*x + 1 38229)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 1/81*(2560*x^3 - 5394*x^2 + 73 92*x - 42385)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 8/81*(748*x^3 + 3795*x^ 2 + 6384*x + 16856)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 43/243*(290*x^3 - 51*x^2 + 1050*x - 1127)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 25/2646*(32* x^3 - 195*x^2 + 60*x - 2408)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 56/81*(1 4*x^3 + 264*x^2 + 168*x + 931)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 49/81* (10*x^3 + 15*x^2 - 42*x + 98)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + 125/162 *(4*x^3 + 6*x^2 + 48*x + 23)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) - 343/243* (2*x^3 + 3*x^2 + 24*x + 133)/(x^4 + 2*x^3 + 15*x^2 + 14*x + 49) + e^x
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx=\frac {x^{7} + 2 \, x^{6} + x^{5} e^{x} + 15 \, x^{5} + 2 \, x^{4} e^{x} + 14 \, x^{4} + 15 \, x^{3} e^{x} + 49 \, x^{3} + 14 \, x^{2} e^{x} + 49 \, x e^{x} - 75}{x^{5} + 2 \, x^{4} + 15 \, x^{3} + 14 \, x^{2} + 49 \, x} \]
integrate(((x^8+3*x^7+24*x^6+43*x^5+168*x^4+147*x^3+343*x^2)*exp(x)+2*x^9+ 6*x^8+48*x^7+86*x^6+336*x^5+294*x^4+686*x^3+375*x^2+225*x+525)/(x^8+3*x^7+ 24*x^6+43*x^5+168*x^4+147*x^3+343*x^2),x, algorithm=\
(x^7 + 2*x^6 + x^5*e^x + 15*x^5 + 2*x^4*e^x + 14*x^4 + 15*x^3*e^x + 49*x^3 + 14*x^2*e^x + 49*x*e^x - 75)/(x^5 + 2*x^4 + 15*x^3 + 14*x^2 + 49*x)
Time = 8.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {525+225 x+375 x^2+686 x^3+294 x^4+336 x^5+86 x^6+48 x^7+6 x^8+2 x^9+e^x \left (343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8\right )}{343 x^2+147 x^3+168 x^4+43 x^5+24 x^6+3 x^7+x^8} \, dx={\mathrm {e}}^x-\frac {75}{x^5+2\,x^4+15\,x^3+14\,x^2+49\,x}+x^2 \]