Integrand size = 27, antiderivative size = 105 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625 (3+2 x)^{9/2}}{1152}-\frac {7925 (3+2 x)^{11/2}}{1408}+\frac {16005 (3+2 x)^{13/2}}{1664}-\frac {17201 (3+2 x)^{15/2}}{1920}+\frac {10475 (3+2 x)^{17/2}}{2176}-\frac {3519 (3+2 x)^{19/2}}{2432}+\frac {27}{128} (3+2 x)^{21/2}-\frac {27 (3+2 x)^{23/2}}{2944} \]
1625/1152*(3+2*x)^(9/2)-7925/1408*(3+2*x)^(11/2)+16005/1664*(3+2*x)^(13/2) -17201/1920*(3+2*x)^(15/2)+10475/2176*(3+2*x)^(17/2)-3519/2432*(3+2*x)^(19 /2)+27/128*(3+2*x)^(21/2)-27/2944*(3+2*x)^(23/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {(3+2 x)^{9/2} \left (-58847566-460865502 x-1481619843 x^2-2481091899 x^3-2232945000 x^4-943203690 x^5-56119635 x^6+56119635 x^7\right )}{47805615} \]
-1/47805615*((3 + 2*x)^(9/2)*(-58847566 - 460865502*x - 1481619843*x^2 - 2 481091899*x^3 - 2232945000*x^4 - 943203690*x^5 - 56119635*x^6 + 56119635*x ^7))
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1195, 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.
| \(\displaystyle \int (5-x) (2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{21/2}+\frac {567}{128} (2 x+3)^{19/2}-\frac {3519}{128} (2 x+3)^{17/2}+\frac {10475}{128} (2 x+3)^{15/2}-\frac {17201}{128} (2 x+3)^{13/2}+\frac {16005}{128} (2 x+3)^{11/2}-\frac {7925}{128} (2 x+3)^{9/2}+\frac {1625}{128} (2 x+3)^{7/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {27 (2 x+3)^{23/2}}{2944}+\frac {27}{128} (2 x+3)^{21/2}-\frac {3519 (2 x+3)^{19/2}}{2432}+\frac {10475 (2 x+3)^{17/2}}{2176}-\frac {17201 (2 x+3)^{15/2}}{1920}+\frac {16005 (2 x+3)^{13/2}}{1664}-\frac {7925 (2 x+3)^{11/2}}{1408}+\frac {1625 (2 x+3)^{9/2}}{1152}\) |
(1625*(3 + 2*x)^(9/2))/1152 - (7925*(3 + 2*x)^(11/2))/1408 + (16005*(3 + 2 *x)^(13/2))/1664 - (17201*(3 + 2*x)^(15/2))/1920 + (10475*(3 + 2*x)^(17/2) )/2176 - (3519*(3 + 2*x)^(19/2))/2432 + (27*(3 + 2*x)^(21/2))/128 - (27*(3 + 2*x)^(23/2))/2944
3.26.42.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.39 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
| method | result | size |
| gosper | \(-\frac {\left (56119635 x^{7}-56119635 x^{6}-943203690 x^{5}-2232945000 x^{4}-2481091899 x^{3}-1481619843 x^{2}-460865502 x -58847566\right ) \left (3+2 x \right )^{\frac {9}{2}}}{47805615}\) | \(45\) |
| pseudoelliptic | \(-\frac {\left (56119635 x^{7}-56119635 x^{6}-943203690 x^{5}-2232945000 x^{4}-2481091899 x^{3}-1481619843 x^{2}-460865502 x -58847566\right ) \left (3+2 x \right )^{\frac {9}{2}}}{47805615}\) | \(45\) |
| trager | \(\left (-\frac {432}{23} x^{11}-\frac {2160}{23} x^{10}+\frac {76392}{437} x^{9}+\frac {19623104}{7429} x^{8}+\frac {1084774681}{111435} x^{7}+\frac {9621056033}{482885} x^{6}+\frac {138248980226}{5311735} x^{5}+\frac {216399786104}{9561123} x^{4}+\frac {41746309645}{3187041} x^{3}+\frac {25807692219}{5311735} x^{2}+\frac {5560131102}{5311735} x +\frac {529628094}{5311735}\right ) \sqrt {3+2 x}\) | \(64\) |
| risch | \(-\frac {\left (897914160 x^{11}+4489570800 x^{10}-8356902840 x^{9}-126274674240 x^{8}-465368338149 x^{7}-952484547267 x^{6}-1244240822034 x^{5}-1081998930520 x^{4}-626194644675 x^{3}-232269229971 x^{2}-50041179918 x -4766652846\right ) \sqrt {3+2 x}}{47805615}\) | \(65\) |
| derivativedivides | \(\frac {1625 \left (3+2 x \right )^{\frac {9}{2}}}{1152}-\frac {7925 \left (3+2 x \right )^{\frac {11}{2}}}{1408}+\frac {16005 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {17201 \left (3+2 x \right )^{\frac {15}{2}}}{1920}+\frac {10475 \left (3+2 x \right )^{\frac {17}{2}}}{2176}-\frac {3519 \left (3+2 x \right )^{\frac {19}{2}}}{2432}+\frac {27 \left (3+2 x \right )^{\frac {21}{2}}}{128}-\frac {27 \left (3+2 x \right )^{\frac {23}{2}}}{2944}\) | \(74\) |
| default | \(\frac {1625 \left (3+2 x \right )^{\frac {9}{2}}}{1152}-\frac {7925 \left (3+2 x \right )^{\frac {11}{2}}}{1408}+\frac {16005 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {17201 \left (3+2 x \right )^{\frac {15}{2}}}{1920}+\frac {10475 \left (3+2 x \right )^{\frac {17}{2}}}{2176}-\frac {3519 \left (3+2 x \right )^{\frac {19}{2}}}{2432}+\frac {27 \left (3+2 x \right )^{\frac {21}{2}}}{128}-\frac {27 \left (3+2 x \right )^{\frac {23}{2}}}{2944}\) | \(74\) |
| meijerg | \(\frac {307481265 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{675675}-\frac {4 \sqrt {\pi }\, \left (-\frac {146432}{729} x^{7}-\frac {259072}{243} x^{6}-\frac {52736}{27} x^{5}-\frac {102400}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{675675}\right )}{256 \sqrt {\pi }}+\frac {375453225 \sqrt {3}\, \left (-\frac {4096 \sqrt {\pi }}{11486475}+\frac {16 \sqrt {\pi }\, \left (\frac {366080}{729} x^{8}+\frac {1903616}{729} x^{7}+\frac {1129216}{243} x^{6}+\frac {25856}{9} x^{5}+\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{11486475}\right )}{256 \sqrt {\pi }}+\frac {204604785 \sqrt {3}\, \left (\frac {8192 \sqrt {\pi }}{43648605}-\frac {8 \sqrt {\pi }\, \left (-\frac {4978688}{2187} x^{9}-\frac {8493056}{729} x^{8}-\frac {14789632}{729} x^{7}-\frac {2951168}{243} x^{6}-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{43648605}\right )}{256 \sqrt {\pi }}+\frac {33297075 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{135135}+\frac {16 \sqrt {\pi }\, \left (\frac {1408}{81} x^{6}+\frac {2560}{27} x^{5}+\frac {14656}{81} x^{4}+\frac {3392}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{135135}\right )}{64 \sqrt {\pi }}+\frac {1862595 \sqrt {3}\, \left (\frac {64 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (-\frac {128}{27} x^{5}-\frac {2176}{81} x^{4}-\frac {1472}{27} x^{3}-\frac {128}{3} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{10395}\right )}{16 \sqrt {\pi }}+\frac {42525 \sqrt {3}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {16 \sqrt {\pi }\, \left (\frac {32}{81} x^{4}+\frac {64}{27} x^{3}+\frac {16}{3} x^{2}+\frac {16}{3} x +2\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{4 \sqrt {\pi }}-\frac {502211745 \sqrt {3}\, \left (\frac {65536 \sqrt {\pi }}{1003917915}-\frac {2 \sqrt {\pi }\, \left (-\frac {1513521152}{19683} x^{11}-\frac {7567605760}{19683} x^{10}-\frac {4261756928}{6561} x^{9}-\frac {271777792}{729} x^{8}-\frac {292864}{729} x^{7}+\frac {157696}{243} x^{6}-\frac {28672}{27} x^{5}+\frac {143360}{81} x^{4}-\frac {81920}{27} x^{3}+\frac {16384}{3} x^{2}-\frac {32768}{3} x +32768\right ) \sqrt {1+\frac {2 x}{3}}}{1003917915}\right )}{4096 \sqrt {\pi }}\) | \(434\) |
-1/47805615*(56119635*x^7-56119635*x^6-943203690*x^5-2232945000*x^4-248109 1899*x^3-1481619843*x^2-460865502*x-58847566)*(3+2*x)^(9/2)
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {1}{47805615} \, {\left (897914160 \, x^{11} + 4489570800 \, x^{10} - 8356902840 \, x^{9} - 126274674240 \, x^{8} - 465368338149 \, x^{7} - 952484547267 \, x^{6} - 1244240822034 \, x^{5} - 1081998930520 \, x^{4} - 626194644675 \, x^{3} - 232269229971 \, x^{2} - 50041179918 \, x - 4766652846\right )} \sqrt {2 \, x + 3} \]
-1/47805615*(897914160*x^11 + 4489570800*x^10 - 8356902840*x^9 - 126274674 240*x^8 - 465368338149*x^7 - 952484547267*x^6 - 1244240822034*x^5 - 108199 8930520*x^4 - 626194644675*x^3 - 232269229971*x^2 - 50041179918*x - 476665 2846)*sqrt(2*x + 3)
Time = 1.53 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=- \frac {27 \left (2 x + 3\right )^{\frac {23}{2}}}{2944} + \frac {27 \left (2 x + 3\right )^{\frac {21}{2}}}{128} - \frac {3519 \left (2 x + 3\right )^{\frac {19}{2}}}{2432} + \frac {10475 \left (2 x + 3\right )^{\frac {17}{2}}}{2176} - \frac {17201 \left (2 x + 3\right )^{\frac {15}{2}}}{1920} + \frac {16005 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} - \frac {7925 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} + \frac {1625 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} \]
-27*(2*x + 3)**(23/2)/2944 + 27*(2*x + 3)**(21/2)/128 - 3519*(2*x + 3)**(1 9/2)/2432 + 10475*(2*x + 3)**(17/2)/2176 - 17201*(2*x + 3)**(15/2)/1920 + 16005*(2*x + 3)**(13/2)/1664 - 7925*(2*x + 3)**(11/2)/1408 + 1625*(2*x + 3 )**(9/2)/1152
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {27}{2944} \, {\left (2 \, x + 3\right )}^{\frac {23}{2}} + \frac {27}{128} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} - \frac {3519}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {10475}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {17201}{1920} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {16005}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {7925}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {1625}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]
-27/2944*(2*x + 3)^(23/2) + 27/128*(2*x + 3)^(21/2) - 3519/2432*(2*x + 3)^ (19/2) + 10475/2176*(2*x + 3)^(17/2) - 17201/1920*(2*x + 3)^(15/2) + 16005 /1664*(2*x + 3)^(13/2) - 7925/1408*(2*x + 3)^(11/2) + 1625/1152*(2*x + 3)^ (9/2)
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {27}{2944} \, {\left (2 \, x + 3\right )}^{\frac {23}{2}} + \frac {27}{128} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} - \frac {3519}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {10475}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {17201}{1920} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {16005}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {7925}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {1625}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]
-27/2944*(2*x + 3)^(23/2) + 27/128*(2*x + 3)^(21/2) - 3519/2432*(2*x + 3)^ (19/2) + 10475/2176*(2*x + 3)^(17/2) - 17201/1920*(2*x + 3)^(15/2) + 16005 /1664*(2*x + 3)^(13/2) - 7925/1408*(2*x + 3)^(11/2) + 1625/1152*(2*x + 3)^ (9/2)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625\,{\left (2\,x+3\right )}^{9/2}}{1152}-\frac {7925\,{\left (2\,x+3\right )}^{11/2}}{1408}+\frac {16005\,{\left (2\,x+3\right )}^{13/2}}{1664}-\frac {17201\,{\left (2\,x+3\right )}^{15/2}}{1920}+\frac {10475\,{\left (2\,x+3\right )}^{17/2}}{2176}-\frac {3519\,{\left (2\,x+3\right )}^{19/2}}{2432}+\frac {27\,{\left (2\,x+3\right )}^{21/2}}{128}-\frac {27\,{\left (2\,x+3\right )}^{23/2}}{2944} \]