Integrand size = 253, antiderivative size = 35 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {e}{2 x \left (x+x \left (3+\frac {e^{2 x}}{x^2-\frac {5}{3+x}}\right )\right )^2} \]
[Out]
\[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {e \left (5-3 x^2-x^3\right ) \left (12 \left (-5+3 x^2+x^3\right )^2+e^{2 x} \left (-45-85 x-29 x^2+30 x^3+23 x^4+4 x^5\right )\right )}{2 x^4 \left (e^{2 x} (3+x)+4 \left (-5+3 x^2+x^3\right )\right )^3} \, dx \\ & = \frac {1}{2} e \int \frac {\left (5-3 x^2-x^3\right ) \left (12 \left (-5+3 x^2+x^3\right )^2+e^{2 x} \left (-45-85 x-29 x^2+30 x^3+23 x^4+4 x^5\right )\right )}{x^4 \left (e^{2 x} (3+x)+4 \left (-5+3 x^2+x^3\right )\right )^3} \, dx \\ & = \frac {1}{2} e \int \left (\frac {8 \left (-5+3 x^2+x^3\right )^2 \left (-35-28 x+6 x^2+10 x^3+2 x^4\right )}{x^3 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {225+425 x+10 x^2-450 x^3-287 x^4+41 x^5+99 x^6+35 x^7+4 x^8}{x^4 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{2} e \int \frac {225+425 x+10 x^2-450 x^3-287 x^4+41 x^5+99 x^6+35 x^7+4 x^8}{x^4 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx\right )+(4 e) \int \frac {\left (-5+3 x^2+x^3\right )^2 \left (-35-28 x+6 x^2+10 x^3+2 x^4\right )}{x^3 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx \\ & = -\left (\frac {1}{2} e \int \left (-\frac {49}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}+\frac {75}{x^4 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}+\frac {350}{3 x^3 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}-\frac {320}{9 x^2 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}-\frac {3730}{27 x \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}+\frac {30 x}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}+\frac {23 x^2}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}+\frac {4 x^3}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}-\frac {50}{27 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2}\right ) \, dx\right )+(4 e) \int \left (\frac {330}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {875}{3 x^3 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {1225}{9 x^2 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}+\frac {12025}{27 x \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {165 x}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {219 x^2}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}-\frac {30 x^3}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}+\frac {36 x^4}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}+\frac {16 x^5}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}+\frac {2 x^6}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}+\frac {125}{27 (3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3}\right ) \, dx \\ & = \frac {1}{27} (25 e) \int \frac {1}{(3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx-(2 e) \int \frac {x^3}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx+(8 e) \int \frac {x^6}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx-\frac {1}{2} (23 e) \int \frac {x^2}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx-(15 e) \int \frac {x}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx+\frac {1}{9} (160 e) \int \frac {1}{x^2 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx+\frac {1}{27} (500 e) \int \frac {1}{(3+x) \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx+\frac {1}{2} (49 e) \int \frac {1}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx-\frac {1}{2} (75 e) \int \frac {1}{x^4 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx-\frac {1}{3} (175 e) \int \frac {1}{x^3 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx+(64 e) \int \frac {x^5}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx+\frac {1}{27} (1865 e) \int \frac {1}{x \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^2} \, dx-(120 e) \int \frac {x^3}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx+(144 e) \int \frac {x^4}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx-\frac {1}{9} (4900 e) \int \frac {1}{x^2 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx-(660 e) \int \frac {x}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx-(876 e) \int \frac {x^2}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx-\frac {1}{3} (3500 e) \int \frac {1}{x^3 \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx+(1320 e) \int \frac {1}{\left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx+\frac {1}{27} (48100 e) \int \frac {1}{x \left (-20+3 e^{2 x}+e^{2 x} x+12 x^2+4 x^3\right )^3} \, dx \\ \end{align*}
Time = 5.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {e \left (-5+3 x^2+x^3\right )^2}{2 x^3 \left (e^{2 x} (3+x)+4 \left (-5+3 x^2+x^3\right )\right )^2} \]
[In]
[Out]
Time = 0.84 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69
method | result | size |
risch | \(\frac {{\mathrm e} \left (x^{6}+6 x^{5}+9 x^{4}-10 x^{3}-30 x^{2}+25\right )}{2 x^{3} \left (x \,{\mathrm e}^{2 x}+4 x^{3}+3 \,{\mathrm e}^{2 x}+12 x^{2}-20\right )^{2}}\) | \(59\) |
parallelrisch | \(\frac {x^{6} {\mathrm e}+6 x^{5} {\mathrm e}+9 x^{4} {\mathrm e}-10 x^{3} {\mathrm e}-30 x^{2} {\mathrm e}+25 \,{\mathrm e}}{2 x^{3} \left (16 x^{6}+8 \,{\mathrm e}^{2 x} x^{4}+x^{2} {\mathrm e}^{4 x}+96 x^{5}+48 \,{\mathrm e}^{2 x} x^{3}+6 x \,{\mathrm e}^{4 x}+144 x^{4}+72 \,{\mathrm e}^{2 x} x^{2}+9 \,{\mathrm e}^{4 x}-160 x^{3}-40 x \,{\mathrm e}^{2 x}-480 x^{2}-120 \,{\mathrm e}^{2 x}+400\right )}\) | \(135\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (33) = 66\).
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {{\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - 10 \, x^{3} - 30 \, x^{2} + 25\right )} e^{3}}{2 \, {\left (16 \, {\left (x^{9} + 6 \, x^{8} + 9 \, x^{7} - 10 \, x^{6} - 30 \, x^{5} + 25 \, x^{3}\right )} e^{2} + {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{\left (4 \, x + 2\right )} + 8 \, {\left (x^{7} + 6 \, x^{6} + 9 \, x^{5} - 5 \, x^{4} - 15 \, x^{3}\right )} e^{\left (2 \, x + 2\right )}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.60 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {e x^{6} + 6 e x^{5} + 9 e x^{4} - 10 e x^{3} - 30 e x^{2} + 25 e}{32 x^{9} + 192 x^{8} + 288 x^{7} - 320 x^{6} - 960 x^{5} + 800 x^{3} + \left (2 x^{5} + 12 x^{4} + 18 x^{3}\right ) e^{4 x} + \left (16 x^{7} + 96 x^{6} + 144 x^{5} - 80 x^{4} - 240 x^{3}\right ) e^{2 x}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (33) = 66\).
Time = 0.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.51 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {x^{6} e + 6 \, x^{5} e + 9 \, x^{4} e - 10 \, x^{3} e - 30 \, x^{2} e + 25 \, e}{2 \, {\left (16 \, x^{9} + 96 \, x^{8} + 144 \, x^{7} - 160 \, x^{6} - 480 \, x^{5} + 400 \, x^{3} + {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{\left (4 \, x\right )} + 8 \, {\left (x^{7} + 6 \, x^{6} + 9 \, x^{5} - 5 \, x^{4} - 15 \, x^{3}\right )} e^{\left (2 \, x\right )}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (33) = 66\).
Time = 0.39 (sec) , antiderivative size = 364, normalized size of antiderivative = 10.40 \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\frac {{\left (2 \, x + 1\right )}^{6} e^{3} + 6 \, {\left (2 \, x + 1\right )}^{5} e^{3} - 9 \, {\left (2 \, x + 1\right )}^{4} e^{3} - 124 \, {\left (2 \, x + 1\right )}^{3} e^{3} - 129 \, {\left (2 \, x + 1\right )}^{2} e^{3} + 630 \, {\left (2 \, x + 1\right )} e^{3} + 1225 \, e^{3}}{4 \, {\left ({\left (2 \, x + 1\right )}^{9} e^{2} + 3 \, {\left (2 \, x + 1\right )}^{8} e^{2} - 24 \, {\left (2 \, x + 1\right )}^{7} e^{2} + 2 \, {\left (2 \, x + 1\right )}^{7} e^{\left (2 \, x + 2\right )} - 80 \, {\left (2 \, x + 1\right )}^{6} e^{2} + 10 \, {\left (2 \, x + 1\right )}^{6} e^{\left (2 \, x + 2\right )} + 210 \, {\left (2 \, x + 1\right )}^{5} e^{2} + {\left (2 \, x + 1\right )}^{5} e^{\left (4 \, x + 2\right )} - 30 \, {\left (2 \, x + 1\right )}^{5} e^{\left (2 \, x + 2\right )} + 654 \, {\left (2 \, x + 1\right )}^{4} e^{2} + 7 \, {\left (2 \, x + 1\right )}^{4} e^{\left (4 \, x + 2\right )} - 150 \, {\left (2 \, x + 1\right )}^{4} e^{\left (2 \, x + 2\right )} - 928 \, {\left (2 \, x + 1\right )}^{3} e^{2} - 2 \, {\left (2 \, x + 1\right )}^{3} e^{\left (4 \, x + 2\right )} + 150 \, {\left (2 \, x + 1\right )}^{3} e^{\left (2 \, x + 2\right )} - 1656 \, {\left (2 \, x + 1\right )}^{2} e^{2} - 46 \, {\left (2 \, x + 1\right )}^{2} e^{\left (4 \, x + 2\right )} + 558 \, {\left (2 \, x + 1\right )}^{2} e^{\left (2 \, x + 2\right )} + 3045 \, {\left (2 \, x + 1\right )} e^{2} + 65 \, {\left (2 \, x + 1\right )} e^{\left (4 \, x + 2\right )} - 890 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x + 2\right )} - 1225 \, e^{2} - 25 \, e^{\left (4 \, x + 2\right )} + 350 \, e^{\left (2 \, x + 2\right )}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{1+2 x} \left (-225-425 x-10 x^2+450 x^3+287 x^4-41 x^5-99 x^6-35 x^7-4 x^8\right )+e \left (1500-2700 x^2-900 x^3+1620 x^4+1080 x^5-144 x^6-324 x^7-108 x^8-12 x^9\right )}{-16000 x^4+28800 x^6+9600 x^7-17280 x^8-11520 x^9+1536 x^{10}+3456 x^{11}+1152 x^{12}+128 x^{13}+e^{6 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^{4 x} \left (-1080 x^4-720 x^5+528 x^6+648 x^7+216 x^8+24 x^9\right )+e^{2 x} \left (7200 x^4+2400 x^5-8640 x^6-5760 x^7+1632 x^8+2592 x^9+864 x^{10}+96 x^{11}\right )} \, dx=\int -\frac {\mathrm {e}\,\left (12\,x^9+108\,x^8+324\,x^7+144\,x^6-1080\,x^5-1620\,x^4+900\,x^3+2700\,x^2-1500\right )+{\mathrm {e}}^{2\,x}\,\mathrm {e}\,\left (4\,x^8+35\,x^7+99\,x^6+41\,x^5-287\,x^4-450\,x^3+10\,x^2+425\,x+225\right )}{{\mathrm {e}}^{4\,x}\,\left (24\,x^9+216\,x^8+648\,x^7+528\,x^6-720\,x^5-1080\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (96\,x^{11}+864\,x^{10}+2592\,x^9+1632\,x^8-5760\,x^7-8640\,x^6+2400\,x^5+7200\,x^4\right )+{\mathrm {e}}^{6\,x}\,\left (2\,x^7+18\,x^6+54\,x^5+54\,x^4\right )-16000\,x^4+28800\,x^6+9600\,x^7-17280\,x^8-11520\,x^9+1536\,x^{10}+3456\,x^{11}+1152\,x^{12}+128\,x^{13}} \,d x \]
[In]
[Out]