Integrand size = 168, antiderivative size = 24 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=\frac {x^4}{-3+\left (4+\frac {4}{\left (1+e^3\right ) x}\right )^2} \] Output:
x^4/(-3+(4+4/(exp(3)+1)/x)^2)
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(24)=48\).
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.88 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=-\frac {7245824+14491648 \left (1+e^3\right ) x+5887232 \left (1+e^3\right )^2 x^2-371293 \left (1+e^3\right )^6 x^6}{371293 \left (1+e^3\right )^4 \left (16+32 \left (1+e^3\right ) x+13 \left (1+e^3\right )^2 x^2\right )} \] Input:
Integrate[(96*x^5 + 160*x^6 + 52*x^7 + 52*E^12*x^7 + E^9*(160*x^6 + 208*x^ 7) + E^3*(192*x^5 + 480*x^6 + 208*x^7) + E^6*(96*x^5 + 480*x^6 + 312*x^7)) /(256 + 1024*x + 1440*x^2 + 832*x^3 + 169*x^4 + 169*E^12*x^4 + E^9*(832*x^ 3 + 676*x^4) + E^3*(1024*x + 2880*x^2 + 2496*x^3 + 676*x^4) + E^6*(1440*x^ 2 + 2496*x^3 + 1014*x^4)),x]
Output:
-1/371293*(7245824 + 14491648*(1 + E^3)*x + 5887232*(1 + E^3)^2*x^2 - 3712 93*(1 + E^3)^6*x^6)/((1 + E^3)^4*(16 + 32*(1 + E^3)*x + 13*(1 + E^3)^2*x^2 ))
Leaf count is larger than twice the leaf count of optimal. \(372\) vs. \(2(24)=48\).
Time = 1.09 (sec) , antiderivative size = 372, normalized size of antiderivative = 15.50, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6, 6, 2459, 1380, 27, 2345, 27, 2019, 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.
| \(\displaystyle \int \frac {52 e^{12} x^7+52 x^7+160 x^6+96 x^5+e^9 \left (208 x^7+160 x^6\right )+e^3 \left (208 x^7+480 x^6+192 x^5\right )+e^6 \left (312 x^7+480 x^6+96 x^5\right )}{169 e^{12} x^4+169 x^4+832 x^3+1440 x^2+e^9 \left (676 x^4+832 x^3\right )+e^3 \left (676 x^4+2496 x^3+2880 x^2+1024 x\right )+e^6 \left (1014 x^4+2496 x^3+1440 x^2\right )+1024 x+256} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {52 e^{12} x^7+52 x^7+160 x^6+96 x^5+e^9 \left (208 x^7+160 x^6\right )+e^3 \left (208 x^7+480 x^6+192 x^5\right )+e^6 \left (312 x^7+480 x^6+96 x^5\right )}{\left (169+169 e^{12}\right ) x^4+832 x^3+1440 x^2+e^9 \left (676 x^4+832 x^3\right )+e^3 \left (676 x^4+2496 x^3+2880 x^2+1024 x\right )+e^6 \left (1014 x^4+2496 x^3+1440 x^2\right )+1024 x+256}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (52+52 e^{12}\right ) x^7+160 x^6+96 x^5+e^9 \left (208 x^7+160 x^6\right )+e^3 \left (208 x^7+480 x^6+192 x^5\right )+e^6 \left (312 x^7+480 x^6+96 x^5\right )}{\left (169+169 e^{12}\right ) x^4+832 x^3+1440 x^2+e^9 \left (676 x^4+832 x^3\right )+e^3 \left (676 x^4+2496 x^3+2880 x^2+1024 x\right )+e^6 \left (1014 x^4+2496 x^3+1440 x^2\right )+1024 x+256}dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {52 \left (1+e^3\right )^4 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^7-288 \left (1+e^3\right )^3 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^6+\frac {7392}{13} \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^5-\frac {58880}{169} \left (1+e^3\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4-\frac {737280 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3}{2197}+\frac {18087936 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{28561 \left (1+e^3\right )}-\frac {127926272 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{371293 \left (1+e^3\right )^2}+\frac {301989888}{4826809 \left (1+e^3\right )^3}}{169 \left (1+e^3\right )^4 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4-96 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2+\frac {2304}{169}}d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle 169 \left (1+e^3\right )^4 \int \frac {4 \left (62748517 \left (1+e^3\right )^4 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^7-347530248 \left (1+e^3\right )^3 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^6+686149464 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^5-420417920 \left (1+e^3\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4-404951040 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3+\frac {764215296 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{1+e^3}-\frac {415760384 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^2}+\frac {75497472}{\left (1+e^3\right )^3}\right )}{4826809 \left (1+e^3\right )^4 \left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )^2}d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \int \frac {62748517 \left (1+e^3\right )^4 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^7-347530248 \left (1+e^3\right )^3 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^6+686149464 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^5-420417920 \left (1+e^3\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4-404951040 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3+\frac {764215296 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{1+e^3}-\frac {415760384 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^2}+\frac {75497472}{\left (1+e^3\right )^3}}{\left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )^2}d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{28561}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {4 \left (-\frac {1}{96} \int \frac {1248 \left (28561 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^5-158184 \left (1+e^3\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4+320424 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3-\frac {236288 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{1+e^3}-\frac {93312 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^2}+\frac {79872}{\left (1+e^3\right )^3}\right )}{48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )-\frac {1024 \left (\frac {17803}{\left (1+e^3\right )^2}-\frac {33150 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{1+e^3}\right )}{13 \left (1+e^3\right )^2 \left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )}\right )}{28561}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (-13 \int \frac {28561 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^5-158184 \left (1+e^3\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4+320424 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3-\frac {236288 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{1+e^3}-\frac {93312 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^2}+\frac {79872}{\left (1+e^3\right )^3}}{48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )-\frac {1024 \left (\frac {17803}{\left (1+e^3\right )^2}-\frac {33150 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{1+e^3}\right )}{13 \left (1+e^3\right )^2 \left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )}\right )}{28561}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \frac {4 \left (-13 \int \left (-169 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3+\left (\frac {936}{\left (1+e^3\right )^2}+\frac {936 e^3}{\left (1+e^3\right )^2}\right ) \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2-\frac {1944 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^2}+\frac {18176}{13 \left (1+e^3\right )^3}+\frac {3456 e^3}{13 \left (1+e^3\right )^4}+\frac {3456}{13 \left (1+e^3\right )^4}\right )d\left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )-\frac {1024 \left (\frac {17803}{\left (1+e^3\right )^2}-\frac {33150 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{1+e^3}\right )}{13 \left (1+e^3\right )^2 \left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )}\right )}{28561}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-\frac {1024 \left (\frac {17803}{\left (1+e^3\right )^2}-\frac {33150 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{1+e^3}\right )}{13 \left (1+e^3\right )^2 \left (48-169 \left (1+e^3\right )^2 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2\right )}-13 \left (-\frac {169}{4} \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^4+\frac {312 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^3}{1+e^3}-\frac {972 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )^2}{\left (1+e^3\right )^2}+\frac {1664 \left (x+\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}\right )}{\left (1+e^3\right )^3}\right )\right )}{28561}\) |
Input:
Int[(96*x^5 + 160*x^6 + 52*x^7 + 52*E^12*x^7 + E^9*(160*x^6 + 208*x^7) + E ^3*(192*x^5 + 480*x^6 + 208*x^7) + E^6*(96*x^5 + 480*x^6 + 312*x^7))/(256 + 1024*x + 1440*x^2 + 832*x^3 + 169*x^4 + 169*E^12*x^4 + E^9*(832*x^3 + 67 6*x^4) + E^3*(1024*x + 2880*x^2 + 2496*x^3 + 676*x^4) + E^6*(1440*x^2 + 24 96*x^3 + 1014*x^4)),x]
Output:
(4*((-1024*(17803/(1 + E^3)^2 - (33150*((832 + 2496*E^3 + 2496*E^6 + 832*E ^9)/(4*(169 + 676*E^3 + 1014*E^6 + 676*E^9 + 169*E^12)) + x))/(1 + E^3)))/ (13*(1 + E^3)^2*(48 - 169*(1 + E^3)^2*((832 + 2496*E^3 + 2496*E^6 + 832*E^ 9)/(4*(169 + 676*E^3 + 1014*E^6 + 676*E^9 + 169*E^12)) + x)^2)) - 13*((166 4*((832 + 2496*E^3 + 2496*E^6 + 832*E^9)/(4*(169 + 676*E^3 + 1014*E^6 + 67 6*E^9 + 169*E^12)) + x))/(1 + E^3)^3 - (972*((832 + 2496*E^3 + 2496*E^6 + 832*E^9)/(4*(169 + 676*E^3 + 1014*E^6 + 676*E^9 + 169*E^12)) + x)^2)/(1 + E^3)^2 + (312*((832 + 2496*E^3 + 2496*E^6 + 832*E^9)/(4*(169 + 676*E^3 + 1 014*E^6 + 676*E^9 + 169*E^12)) + x)^3)/(1 + E^3) - (169*((832 + 2496*E^3 + 2496*E^6 + 832*E^9)/(4*(169 + 676*E^3 + 1014*E^6 + 676*E^9 + 169*E^12)) + x)^4)/4)))/28561
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 0.48 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00
| method | result | size |
| gosper | \(\frac {x^{6} \left (2 \,{\mathrm e}^{3}+{\mathrm e}^{6}+1\right )}{26 x^{2} {\mathrm e}^{3}+13 x^{2} {\mathrm e}^{6}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\) | \(48\) |
| norman | \(\frac {x^{6} \left (2 \,{\mathrm e}^{3}+{\mathrm e}^{6}+1\right )}{26 x^{2} {\mathrm e}^{3}+13 x^{2} {\mathrm e}^{6}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\) | \(48\) |
| parallelrisch | \(\frac {16 \,{\mathrm e}^{6} x^{6}+32 x^{6} {\mathrm e}^{3}+16 x^{6}}{416 x^{2} {\mathrm e}^{3}+208 x^{2} {\mathrm e}^{6}+512 x \,{\mathrm e}^{3}+208 x^{2}+512 x +256}\) | \(58\) |
| risch | \(\frac {-7245824-14491648 x +2227758 x^{6} {\mathrm e}^{3}-5887232 x^{2} {\mathrm e}^{6}-11774464 x^{2} {\mathrm e}^{3}-14491648 x \,{\mathrm e}^{3}+371293 x^{6}-5887232 x^{2}+371293 x^{6} {\mathrm e}^{18}+5569395 \,{\mathrm e}^{6} x^{6}+5569395 \,{\mathrm e}^{12} x^{6}+7425860 \,{\mathrm e}^{9} x^{6}+2227758 x^{6} {\mathrm e}^{15}}{371293 \left ({\mathrm e}^{3}+1\right )^{4} \left (26 x^{2} {\mathrm e}^{3}+13 x^{2} {\mathrm e}^{6}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16\right )}\) | \(116\) |
| default | \(\text {Expression too large to display}\) | \(2908\) |
Input:
int((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*e xp(3)^2+(208*x^7+480*x^6+192*x^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*e xp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp(3)^2+( 676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+2 56),x,method=_RETURNVERBOSE)
Output:
x^6*(exp(3)^2+2*exp(3)+1)/(13*x^2*exp(3)^2+26*x^2*exp(3)+32*x*exp(3)+13*x^ 2+32*x+16)
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 6.83 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=\frac {371293 \, x^{6} e^{18} + 2227758 \, x^{6} e^{15} + 5569395 \, x^{6} e^{12} + 7425860 \, x^{6} e^{9} + 371293 \, x^{6} - 5887232 \, x^{2} + 13 \, {\left (428415 \, x^{6} - 452864 \, x^{2}\right )} e^{6} + 2 \, {\left (1113879 \, x^{6} - 5887232 \, x^{2} - 7245824 \, x\right )} e^{3} - 14491648 \, x - 7245824}{371293 \, {\left (13 \, x^{2} e^{18} + 13 \, x^{2} + 2 \, {\left (39 \, x^{2} + 16 \, x\right )} e^{15} + {\left (195 \, x^{2} + 160 \, x + 16\right )} e^{12} + 4 \, {\left (65 \, x^{2} + 80 \, x + 16\right )} e^{9} + {\left (195 \, x^{2} + 320 \, x + 96\right )} e^{6} + 2 \, {\left (39 \, x^{2} + 80 \, x + 32\right )} e^{3} + 32 \, x + 16\right )}} \] Input:
integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96* x^5)*exp(3)^2+(208*x^7+480*x^6+192*x^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169 *x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp( 3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+10 24*x+256),x, algorithm="fricas")
Output:
1/371293*(371293*x^6*e^18 + 2227758*x^6*e^15 + 5569395*x^6*e^12 + 7425860* x^6*e^9 + 371293*x^6 - 5887232*x^2 + 13*(428415*x^6 - 452864*x^2)*e^6 + 2* (1113879*x^6 - 5887232*x^2 - 7245824*x)*e^3 - 14491648*x - 7245824)/(13*x^ 2*e^18 + 13*x^2 + 2*(39*x^2 + 16*x)*e^15 + (195*x^2 + 160*x + 16)*e^12 + 4 *(65*x^2 + 80*x + 16)*e^9 + (195*x^2 + 320*x + 96)*e^6 + 2*(39*x^2 + 80*x + 32)*e^3 + 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (15) = 30\).
Time = 2.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.29 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=\frac {x^{4}}{13} - \frac {32 x^{3}}{169 + 169 e^{3}} + \frac {816 x^{2}}{2197 + 4394 e^{3} + 2197 e^{6}} - \frac {19456 x}{28561 + 85683 e^{3} + 85683 e^{6} + 28561 e^{9}} + \frac {x \left (- 10444800 e^{3} - 10444800\right ) - 7245824}{x^{2} \cdot \left (4826809 + 28960854 e^{3} + 72402135 e^{6} + 96536180 e^{9} + 72402135 e^{12} + 28960854 e^{15} + 4826809 e^{18}\right ) + x \left (11881376 + 59406880 e^{3} + 118813760 e^{6} + 118813760 e^{9} + 59406880 e^{12} + 11881376 e^{15}\right ) + 5940688 + 23762752 e^{3} + 35644128 e^{6} + 23762752 e^{9} + 5940688 e^{12}} \] Input:
integrate((52*x**7*exp(3)**4+(208*x**7+160*x**6)*exp(3)**3+(312*x**7+480*x **6+96*x**5)*exp(3)**2+(208*x**7+480*x**6+192*x**5)*exp(3)+52*x**7+160*x** 6+96*x**5)/(169*x**4*exp(3)**4+(676*x**4+832*x**3)*exp(3)**3+(1014*x**4+24 96*x**3+1440*x**2)*exp(3)**2+(676*x**4+2496*x**3+2880*x**2+1024*x)*exp(3)+ 169*x**4+832*x**3+1440*x**2+1024*x+256),x)
Output:
x**4/13 - 32*x**3/(169 + 169*exp(3)) + 816*x**2/(2197 + 4394*exp(3) + 2197 *exp(6)) - 19456*x/(28561 + 85683*exp(3) + 85683*exp(6) + 28561*exp(9)) + (x*(-10444800*exp(3) - 10444800) - 7245824)/(x**2*(4826809 + 28960854*exp( 3) + 72402135*exp(6) + 96536180*exp(9) + 72402135*exp(12) + 28960854*exp(1 5) + 4826809*exp(18)) + x*(11881376 + 59406880*exp(3) + 118813760*exp(6) + 118813760*exp(9) + 59406880*exp(12) + 11881376*exp(15)) + 5940688 + 23762 752*exp(3) + 35644128*exp(6) + 23762752*exp(9) + 5940688*exp(12))
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (24) = 48\).
Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.96 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=-\frac {4096 \, {\left (2550 \, x {\left (e^{3} + 1\right )} + 1769\right )}}{371293 \, {\left (13 \, x^{2} {\left (e^{18} + 6 \, e^{15} + 15 \, e^{12} + 20 \, e^{9} + 15 \, e^{6} + 6 \, e^{3} + 1\right )} + 32 \, x {\left (e^{15} + 5 \, e^{12} + 10 \, e^{9} + 10 \, e^{6} + 5 \, e^{3} + 1\right )} + 16 \, e^{12} + 64 \, e^{9} + 96 \, e^{6} + 64 \, e^{3} + 16\right )}} + \frac {2197 \, x^{4} {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )} - 5408 \, x^{3} {\left (e^{6} + 2 \, e^{3} + 1\right )} + 10608 \, x^{2} {\left (e^{3} + 1\right )} - 19456 \, x}{28561 \, {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )}} \] Input:
integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96* x^5)*exp(3)^2+(208*x^7+480*x^6+192*x^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169 *x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp( 3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+10 24*x+256),x, algorithm="maxima")
Output:
-4096/371293*(2550*x*(e^3 + 1) + 1769)/(13*x^2*(e^18 + 6*e^15 + 15*e^12 + 20*e^9 + 15*e^6 + 6*e^3 + 1) + 32*x*(e^15 + 5*e^12 + 10*e^9 + 10*e^6 + 5*e ^3 + 1) + 16*e^12 + 64*e^9 + 96*e^6 + 64*e^3 + 16) + 1/28561*(2197*x^4*(e^ 9 + 3*e^6 + 3*e^3 + 1) - 5408*x^3*(e^6 + 2*e^3 + 1) + 10608*x^2*(e^3 + 1) - 19456*x)/(e^9 + 3*e^6 + 3*e^3 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 529, normalized size of antiderivative = 22.04 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96* x^5)*exp(3)^2+(208*x^7+480*x^6+192*x^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169 *x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp( 3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+10 24*x+256),x, algorithm="giac")
Output:
1/28561*(2197*x^4*e^48 + 35152*x^4*e^45 + 263640*x^4*e^42 + 1230320*x^4*e^ 39 + 3998540*x^4*e^36 + 9596496*x^4*e^33 + 17593576*x^4*e^30 + 25133680*x^ 4*e^27 + 28275390*x^4*e^24 + 25133680*x^4*e^21 + 17593576*x^4*e^18 + 95964 96*x^4*e^15 + 3998540*x^4*e^12 + 1230320*x^4*e^9 + 263640*x^4*e^6 + 35152* x^4*e^3 + 2197*x^4 - 5408*x^3*e^45 - 81120*x^3*e^42 - 567840*x^3*e^39 - 24 60640*x^3*e^36 - 7381920*x^3*e^33 - 16240224*x^3*e^30 - 27067040*x^3*e^27 - 34800480*x^3*e^24 - 34800480*x^3*e^21 - 27067040*x^3*e^18 - 16240224*x^3 *e^15 - 7381920*x^3*e^12 - 2460640*x^3*e^9 - 567840*x^3*e^6 - 81120*x^3*e^ 3 - 5408*x^3 + 10608*x^2*e^42 + 148512*x^2*e^39 + 965328*x^2*e^36 + 386131 2*x^2*e^33 + 10618608*x^2*e^30 + 21237216*x^2*e^27 + 31855824*x^2*e^24 + 3 6406656*x^2*e^21 + 31855824*x^2*e^18 + 21237216*x^2*e^15 + 10618608*x^2*e^ 12 + 3861312*x^2*e^9 + 965328*x^2*e^6 + 148512*x^2*e^3 + 10608*x^2 - 19456 *x*e^39 - 252928*x*e^36 - 1517568*x*e^33 - 5564416*x*e^30 - 13911040*x*e^2 7 - 25039872*x*e^24 - 33386496*x*e^21 - 33386496*x*e^18 - 25039872*x*e^15 - 13911040*x*e^12 - 5564416*x*e^9 - 1517568*x*e^6 - 252928*x*e^3 - 19456*x )/(e^48 + 16*e^45 + 120*e^42 + 560*e^39 + 1820*e^36 + 4368*e^33 + 8008*e^3 0 + 11440*e^27 + 12870*e^24 + 11440*e^21 + 8008*e^18 + 4368*e^15 + 1820*e^ 12 + 560*e^9 + 120*e^6 + 16*e^3 + 1) - 4096/371293*(2550*x*e^3 + 2550*x + 1769)/((13*x^2*e^6 + 26*x^2*e^3 + 13*x^2 + 32*x*e^3 + 32*x + 16)*(e^12 + 4 *e^9 + 6*e^6 + 4*e^3 + 1))
Time = 3.93 (sec) , antiderivative size = 133, normalized size of antiderivative = 5.54 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=\frac {816\,x^2}{2197\,{\left ({\mathrm {e}}^3+1\right )}^2}-\frac {32\,x^3}{169\,\left ({\mathrm {e}}^3+1\right )}+x\,\left (\frac {33792}{28561\,{\left ({\mathrm {e}}^3+1\right )}^3}-\frac {4\,\left (1024\,{\mathrm {e}}^3+1024\right )}{2197\,{\left ({\mathrm {e}}^3+1\right )}^4}\right )+\frac {x^4}{13}-\frac {\frac {10444800\,x}{13}+\frac {7245824}{13\,\left ({\mathrm {e}}^3+1\right )}}{\left (1856465\,{\mathrm {e}}^3+3712930\,{\mathrm {e}}^6+3712930\,{\mathrm {e}}^9+1856465\,{\mathrm {e}}^{12}+371293\,{\mathrm {e}}^{15}+371293\right )\,x^2+\left (3655808\,{\mathrm {e}}^3+5483712\,{\mathrm {e}}^6+3655808\,{\mathrm {e}}^9+913952\,{\mathrm {e}}^{12}+913952\right )\,x+1370928\,{\mathrm {e}}^3+1370928\,{\mathrm {e}}^6+456976\,{\mathrm {e}}^9+456976} \] Input:
int((exp(9)*(160*x^6 + 208*x^7) + 52*x^7*exp(12) + exp(3)*(192*x^5 + 480*x ^6 + 208*x^7) + exp(6)*(96*x^5 + 480*x^6 + 312*x^7) + 96*x^5 + 160*x^6 + 5 2*x^7)/(1024*x + exp(9)*(832*x^3 + 676*x^4) + 169*x^4*exp(12) + exp(3)*(10 24*x + 2880*x^2 + 2496*x^3 + 676*x^4) + exp(6)*(1440*x^2 + 2496*x^3 + 1014 *x^4) + 1440*x^2 + 832*x^3 + 169*x^4 + 256),x)
Output:
(816*x^2)/(2197*(exp(3) + 1)^2) - (32*x^3)/(169*(exp(3) + 1)) + x*(33792/( 28561*(exp(3) + 1)^3) - (4*(1024*exp(3) + 1024))/(2197*(exp(3) + 1)^4)) + x^4/13 - ((10444800*x)/13 + 7245824/(13*(exp(3) + 1)))/(1370928*exp(3) + 1 370928*exp(6) + 456976*exp(9) + x*(3655808*exp(3) + 5483712*exp(6) + 36558 08*exp(9) + 913952*exp(12) + 913952) + x^2*(1856465*exp(3) + 3712930*exp(6 ) + 3712930*exp(9) + 1856465*exp(12) + 371293*exp(15) + 371293) + 456976)
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx=\frac {x^{6} \left (e^{6}+2 e^{3}+1\right )}{13 e^{6} x^{2}+26 e^{3} x^{2}+32 e^{3} x +13 x^{2}+32 x +16} \] Input:
int((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*e xp(3)^2+(208*x^7+480*x^6+192*x^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*e xp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp(3)^2+( 676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+2 56),x)
Output:
(x**6*(e**6 + 2*e**3 + 1))/(13*e**6*x**2 + 26*e**3*x**2 + 32*e**3*x + 13*x **2 + 32*x + 16)