Integrand size = 27, antiderivative size = 254 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=-\frac {636602271789 (1-4 x) \sqrt {3-x+2 x^2}}{34359738368}-\frac {9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac {401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac {25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac {112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac {122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac {23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac {3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac {1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac {13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac {625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}-\frac {14641852251147 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{68719476736 \sqrt {2}} \]
-9226119881/2147483648*(1-4*x)*(2*x^2-x+3)^(3/2)-401135647/335544320*(1-4* x)*(2*x^2-x+3)^(5/2)+25250178739/5725224960*(2*x^2-x+3)^(7/2)+112244125/12 2683392*x*(2*x^2-x+3)^(7/2)+122595067/19169280*x^2*(2*x^2-x+3)^(7/2)+23460 839/532480*x^3*(2*x^2-x+3)^(7/2)+3684995/39936*x^4*(2*x^2-x+3)^(7/2)+10462 25/9984*x^5*(2*x^2-x+3)^(7/2)+13875/208*x^6*(2*x^2-x+3)^(7/2)+625/28*x^7*( 2*x^2-x+3)^(7/2)-14641852251147/137438953472*arcsinh(1/23*(1-4*x)*23^(1/2) )*2^(1/2)-636602271789/34359738368*(1-4*x)*(2*x^2-x+3)^(1/2)
Time = 1.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.45 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (10820567498568669+12071614275862524 x+50064174038215008 x^2+142490931553577856 x^3+257786732552566784 x^4+405468382284161024 x^5+485091164642279424 x^6+530502956133122048 x^7+439064558846345216 x^8+363646430503501824 x^9+204932411660697600 x^{10}+137233466130432000 x^{11}+37398427729920000 x^{12}+25125558681600000 x^{13}\right )-59958384968446965 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{562812514467840} \]
(4*Sqrt[3 - x + 2*x^2]*(10820567498568669 + 12071614275862524*x + 50064174 038215008*x^2 + 142490931553577856*x^3 + 257786732552566784*x^4 + 40546838 2284161024*x^5 + 485091164642279424*x^6 + 530502956133122048*x^7 + 4390645 58846345216*x^8 + 363646430503501824*x^9 + 204932411660697600*x^10 + 13723 3466130432000*x^11 + 37398427729920000*x^12 + 25125558681600000*x^13) - 59 958384968446965*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/5628125144 67840
Time = 0.86 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.20, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1087, 1087, 1090, 222}
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.
| \(\displaystyle \int \left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )^4 \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{28} \int \frac {7}{2} \left (2 x^2-x+3\right )^{5/2} \left (13875 x^7+15050 x^6+18720 x^5+14088 x^4+7488 x^3+3008 x^2+768 x+128\right )dx+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \left (2 x^2-x+3\right )^{5/2} \left (13875 x^7+15050 x^6+18720 x^5+14088 x^4+7488 x^3+3008 x^2+768 x+128\right )dx+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{26} \int \frac {1}{2} \left (2 x^2-x+3\right )^{5/2} \left (1046225 x^6+473940 x^5+732576 x^4+389376 x^3+156416 x^2+39936 x+6656\right )dx+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \int \left (2 x^2-x+3\right )^{5/2} \left (1046225 x^6+473940 x^5+732576 x^4+389376 x^3+156416 x^2+39936 x+6656\right )dx+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{24} \int \frac {1}{2} \left (2 x^2-x+3\right )^{5/2} \left (40534945 x^5+3776898 x^4+18690048 x^3+7507968 x^2+1916928 x+319488\right )dx+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \int \left (2 x^2-x+3\right )^{5/2} \left (40534945 x^5+3776898 x^4+18690048 x^3+7507968 x^2+1916928 x+319488\right )dx+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {1}{22} \int \frac {33}{2} \left (2 x^2-x+3\right )^{5/2} \left (23460839 x^4-4559896 x^3+10010624 x^2+2555904 x+425984\right )dx+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \int \left (2 x^2-x+3\right )^{5/2} \left (23460839 x^4-4559896 x^3+10010624 x^2+2555904 x+425984\right )dx+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{20} \int \frac {1}{2} \left (2 x^2-x+3\right )^{5/2} \left (122595067 x^3-21870142 x^2+102236160 x+17039360\right )dx+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \int \left (2 x^2-x+3\right )^{5/2} \left (122595067 x^3-21870142 x^2+102236160 x+17039360\right )dx+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{18} \int \frac {1}{2} \left (2 x^2-x+3\right )^{5/2} \left (561220625 x^2+2209360956 x+613416960\right )dx+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \int \left (2 x^2-x+3\right )^{5/2} \left (561220625 x^2+2209360956 x+613416960\right )dx+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {1}{16} \int \frac {3}{2} (25250178739 x+5420672990) \left (2 x^2-x+3\right )^{5/2}dx+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \int (25250178739 x+5420672990) \left (2 x^2-x+3\right )^{5/2}dx+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \int \left (2 x^2-x+3\right )^{5/2}dx+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \left (\frac {115}{48} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \left (\frac {115}{48} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{52} \left (\frac {1}{48} \left (\frac {3}{4} \left (\frac {1}{40} \left (\frac {1}{36} \left (\frac {3}{32} \left (\frac {46932870699}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25250178739}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {561220625}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {122595067}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {23460839}{20} x^3 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {3684995}{2} \left (2 x^2-x+3\right )^{7/2} x^4\right )+\frac {1046225}{24} \left (2 x^2-x+3\right )^{7/2} x^5\right )+\frac {13875}{26} \left (2 x^2-x+3\right )^{7/2} x^6\right )+\frac {625}{28} \left (2 x^2-x+3\right )^{7/2} x^7\) |
(625*x^7*(3 - x + 2*x^2)^(7/2))/28 + ((13875*x^6*(3 - x + 2*x^2)^(7/2))/26 + ((1046225*x^5*(3 - x + 2*x^2)^(7/2))/24 + ((3684995*x^4*(3 - x + 2*x^2) ^(7/2))/2 + (3*((23460839*x^3*(3 - x + 2*x^2)^(7/2))/20 + ((122595067*x^2* (3 - x + 2*x^2)^(7/2))/18 + ((561220625*x*(3 - x + 2*x^2)^(7/2))/16 + (3*( (25250178739*(3 - x + 2*x^2)^(7/2))/14 + (46932870699*(-1/24*((1 - 4*x)*(3 - x + 2*x^2)^(5/2)) + (115*(-1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69 *(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]]) /(16*Sqrt[2])))/32))/48))/4))/32)/36)/40))/4)/48)/52)/8
3.1.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 0.84 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(\frac {\left (25125558681600000 x^{13}+37398427729920000 x^{12}+137233466130432000 x^{11}+204932411660697600 x^{10}+363646430503501824 x^{9}+439064558846345216 x^{8}+530502956133122048 x^{7}+485091164642279424 x^{6}+405468382284161024 x^{5}+257786732552566784 x^{4}+142490931553577856 x^{3}+50064174038215008 x^{2}+12071614275862524 x +10820567498568669\right ) \sqrt {2 x^{2}-x +3}}{140703128616960}+\frac {14641852251147 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{137438953472}\) | \(95\) |
| trager | \(\left (\frac {1250}{7} x^{13}+\frac {48375}{182} x^{12}+\frac {1217225}{1248} x^{11}+\frac {50895515}{34944} x^{10}+\frac {172023939}{66560} x^{9}+\frac {52340574127}{16773120} x^{8}+\frac {2023708176167}{536739840} x^{7}+\frac {2467301252453}{715653120} x^{6}+\frac {49495652134297}{17175674880} x^{5}+\frac {17981775429169}{9814671360} x^{4}+\frac {371070134254109}{366414397440} x^{3}+\frac {24833419661813}{69793218560} x^{2}+\frac {335322618773959}{3908420239360} x +\frac {1202285277618741}{15633680957440}\right ) \sqrt {2 x^{2}-x +3}+\frac {14641852251147 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{137438953472}\) | \(121\) |
| default | \(\frac {401135647 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{335544320}+\frac {9226119881 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{2147483648}+\frac {636602271789 \left (-1+4 x \right ) \sqrt {2 x^{2}-x +3}}{34359738368}+\frac {14641852251147 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{137438953472}+\frac {25250178739 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{5725224960}+\frac {625 x^{7} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{28}+\frac {13875 x^{6} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{208}+\frac {1046225 x^{5} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{9984}+\frac {3684995 x^{4} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{39936}+\frac {23460839 x^{3} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{532480}+\frac {122595067 x^{2} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{19169280}+\frac {112244125 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{122683392}\) | \(204\) |
1/140703128616960*(25125558681600000*x^13+37398427729920000*x^12+137233466 130432000*x^11+204932411660697600*x^10+363646430503501824*x^9+439064558846 345216*x^8+530502956133122048*x^7+485091164642279424*x^6+40546838228416102 4*x^5+257786732552566784*x^4+142490931553577856*x^3+50064174038215008*x^2+ 12071614275862524*x+10820567498568669)*(2*x^2-x+3)^(1/2)+14641852251147/13 7438953472*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.46 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{140703128616960} \, {\left (25125558681600000 \, x^{13} + 37398427729920000 \, x^{12} + 137233466130432000 \, x^{11} + 204932411660697600 \, x^{10} + 363646430503501824 \, x^{9} + 439064558846345216 \, x^{8} + 530502956133122048 \, x^{7} + 485091164642279424 \, x^{6} + 405468382284161024 \, x^{5} + 257786732552566784 \, x^{4} + 142490931553577856 \, x^{3} + 50064174038215008 \, x^{2} + 12071614275862524 \, x + 10820567498568669\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {14641852251147}{274877906944} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/140703128616960*(25125558681600000*x^13 + 37398427729920000*x^12 + 13723 3466130432000*x^11 + 204932411660697600*x^10 + 363646430503501824*x^9 + 43 9064558846345216*x^8 + 530502956133122048*x^7 + 485091164642279424*x^6 + 4 05468382284161024*x^5 + 257786732552566784*x^4 + 142490931553577856*x^3 + 50064174038215008*x^2 + 12071614275862524*x + 10820567498568669)*sqrt(2*x^ 2 - x + 3) + 14641852251147/274877906944*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.90 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.49 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {1250 x^{13}}{7} + \frac {48375 x^{12}}{182} + \frac {1217225 x^{11}}{1248} + \frac {50895515 x^{10}}{34944} + \frac {172023939 x^{9}}{66560} + \frac {52340574127 x^{8}}{16773120} + \frac {2023708176167 x^{7}}{536739840} + \frac {2467301252453 x^{6}}{715653120} + \frac {49495652134297 x^{5}}{17175674880} + \frac {17981775429169 x^{4}}{9814671360} + \frac {371070134254109 x^{3}}{366414397440} + \frac {24833419661813 x^{2}}{69793218560} + \frac {335322618773959 x}{3908420239360} + \frac {1202285277618741}{15633680957440}\right ) + \frac {14641852251147 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{137438953472} \]
sqrt(2*x**2 - x + 3)*(1250*x**13/7 + 48375*x**12/182 + 1217225*x**11/1248 + 50895515*x**10/34944 + 172023939*x**9/66560 + 52340574127*x**8/16773120 + 2023708176167*x**7/536739840 + 2467301252453*x**6/715653120 + 4949565213 4297*x**5/17175674880 + 17981775429169*x**4/9814671360 + 371070134254109*x **3/366414397440 + 24833419661813*x**2/69793218560 + 335322618773959*x/390 8420239360 + 1202285277618741/15633680957440) + 14641852251147*sqrt(2)*asi nh(4*sqrt(23)*(x - 1/4)/23)/137438953472
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.93 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {625}{28} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{7} + \frac {13875}{208} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{6} + \frac {1046225}{9984} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{5} + \frac {3684995}{39936} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{4} + \frac {23460839}{532480} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{3} + \frac {122595067}{19169280} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} + \frac {112244125}{122683392} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {25250178739}{5725224960} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {401135647}{83886080} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {401135647}{335544320} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {9226119881}{536870912} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {9226119881}{2147483648} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {636602271789}{8589934592} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {14641852251147}{137438953472} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {636602271789}{34359738368} \, \sqrt {2 \, x^{2} - x + 3} \]
625/28*(2*x^2 - x + 3)^(7/2)*x^7 + 13875/208*(2*x^2 - x + 3)^(7/2)*x^6 + 1 046225/9984*(2*x^2 - x + 3)^(7/2)*x^5 + 3684995/39936*(2*x^2 - x + 3)^(7/2 )*x^4 + 23460839/532480*(2*x^2 - x + 3)^(7/2)*x^3 + 122595067/19169280*(2* x^2 - x + 3)^(7/2)*x^2 + 112244125/122683392*(2*x^2 - x + 3)^(7/2)*x + 252 50178739/5725224960*(2*x^2 - x + 3)^(7/2) + 401135647/83886080*(2*x^2 - x + 3)^(5/2)*x - 401135647/335544320*(2*x^2 - x + 3)^(5/2) + 9226119881/5368 70912*(2*x^2 - x + 3)^(3/2)*x - 9226119881/2147483648*(2*x^2 - x + 3)^(3/2 ) + 636602271789/8589934592*sqrt(2*x^2 - x + 3)*x + 14641852251147/1374389 53472*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 636602271789/34359738368* sqrt(2*x^2 - x + 3)
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.44 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{140703128616960} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (32 \, {\left (12 \, {\left (200 \, {\left (20 \, {\left (240 \, {\left (260 \, x + 387\right )} x + 340823\right )} x + 10179103\right )} x + 3612502719\right )} x + 52340574127\right )} x + 2023708176167\right )} x + 7401903757359\right )} x + 49495652134297\right )} x + 125872428004183\right )} x + 1113210402762327\right )} x + 1564505438694219\right )} x + 3017903568965631\right )} x + 10820567498568669\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {14641852251147}{137438953472} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/140703128616960*(4*(8*(4*(16*(4*(8*(4*(32*(12*(200*(20*(240*(260*x + 387 )*x + 340823)*x + 10179103)*x + 3612502719)*x + 52340574127)*x + 202370817 6167)*x + 7401903757359)*x + 49495652134297)*x + 125872428004183)*x + 1113 210402762327)*x + 1564505438694219)*x + 3017903568965631)*x + 108205674985 68669)*sqrt(2*x^2 - x + 3) - 14641852251147/137438953472*sqrt(2)*log(-2*sq rt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx=\int {\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^4 \,d x \]