Integrand size = 27, antiderivative size = 105 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625}{384} (3+2 x)^{3/2}-\frac {1585}{128} (3+2 x)^{5/2}+\frac {16005}{896} (3+2 x)^{7/2}-\frac {17201 (3+2 x)^{9/2}}{1152}+\frac {10475 (3+2 x)^{11/2}}{1408}-\frac {3519 (3+2 x)^{13/2}}{1664}+\frac {189}{640} (3+2 x)^{15/2}-\frac {27 (3+2 x)^{17/2}}{2176} \]
1625/384*(3+2*x)^(3/2)-1585/128*(3+2*x)^(5/2)+16005/896*(3+2*x)^(7/2)-1720 1/1152*(3+2*x)^(9/2)+10475/1408*(3+2*x)^(11/2)-3519/1664*(3+2*x)^(13/2)+18 9/640*(3+2*x)^(15/2)-27/2176*(3+2*x)^(17/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {(3+2 x)^{3/2} \left (-1197186-9013014 x-29756385 x^2-50880095 x^3-47286540 x^4-20968794 x^5-1702701 x^6+1216215 x^7\right )}{765765} \]
-1/765765*((3 + 2*x)^(3/2)*(-1197186 - 9013014*x - 29756385*x^2 - 50880095 *x^3 - 47286540*x^4 - 20968794*x^5 - 1702701*x^6 + 1216215*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.
\(\displaystyle \int (5-x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^3 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{15/2}+\frac {567}{128} (2 x+3)^{13/2}-\frac {3519}{128} (2 x+3)^{11/2}+\frac {10475}{128} (2 x+3)^{9/2}-\frac {17201}{128} (2 x+3)^{7/2}+\frac {16005}{128} (2 x+3)^{5/2}-\frac {7925}{128} (2 x+3)^{3/2}+\frac {1625}{128} \sqrt {2 x+3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {27 (2 x+3)^{17/2}}{2176}+\frac {189}{640} (2 x+3)^{15/2}-\frac {3519 (2 x+3)^{13/2}}{1664}+\frac {10475 (2 x+3)^{11/2}}{1408}-\frac {17201 (2 x+3)^{9/2}}{1152}+\frac {16005}{896} (2 x+3)^{7/2}-\frac {1585}{128} (2 x+3)^{5/2}+\frac {1625}{384} (2 x+3)^{3/2}\) |
(1625*(3 + 2*x)^(3/2))/384 - (1585*(3 + 2*x)^(5/2))/128 + (16005*(3 + 2*x) ^(7/2))/896 - (17201*(3 + 2*x)^(9/2))/1152 + (10475*(3 + 2*x)^(11/2))/1408 - (3519*(3 + 2*x)^(13/2))/1664 + (189*(3 + 2*x)^(15/2))/640 - (27*(3 + 2* x)^(17/2))/2176
3.26.45.3.1 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 = 0.38 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (1216215 x^{7}-1702701 x^{6}-20968794 x^{5}-47286540 x^{4}-50880095 x^{3}-29756385 x^{2}-9013014 x -1197186\right ) \left (3+2 x \right )^{\frac {3}{2}}}{765765}\) | \(45\) |
trager | \(\left (-\frac {54}{17} x^{8}-\frac {27}{85} x^{7}+\frac {67887}{1105} x^{6}+\frac {2499674}{12155} x^{5}+\frac {6960566}{21879} x^{4}+\frac {14143537}{51051} x^{3}+\frac {11921687}{85085} x^{2}+\frac {9811138}{255255} x +\frac {399062}{85085}\right ) \sqrt {3+2 x}\) | \(49\) |
risch | \(-\frac {\left (2432430 x^{8}+243243 x^{7}-47045691 x^{6}-157479462 x^{5}-243619810 x^{4}-212153055 x^{3}-107295183 x^{2}-29433414 x -3591558\right ) \sqrt {3+2 x}}{765765}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (2432430 x^{8}+243243 x^{7}-47045691 x^{6}-157479462 x^{5}-243619810 x^{4}-212153055 x^{3}-107295183 x^{2}-29433414 x -3591558\right ) \sqrt {3+2 x}}{765765}\) | \(50\) |
derivativedivides | \(\frac {1625 \left (3+2 x \right )^{\frac {3}{2}}}{384}-\frac {1585 \left (3+2 x \right )^{\frac {5}{2}}}{128}+\frac {16005 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {17201 \left (3+2 x \right )^{\frac {9}{2}}}{1152}+\frac {10475 \left (3+2 x \right )^{\frac {11}{2}}}{1408}-\frac {3519 \left (3+2 x \right )^{\frac {13}{2}}}{1664}+\frac {189 \left (3+2 x \right )^{\frac {15}{2}}}{640}-\frac {27 \left (3+2 x \right )^{\frac {17}{2}}}{2176}\) | \(74\) |
default | \(\frac {1625 \left (3+2 x \right )^{\frac {3}{2}}}{384}-\frac {1585 \left (3+2 x \right )^{\frac {5}{2}}}{128}+\frac {16005 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {17201 \left (3+2 x \right )^{\frac {9}{2}}}{1152}+\frac {10475 \left (3+2 x \right )^{\frac {11}{2}}}{1408}-\frac {3519 \left (3+2 x \right )^{\frac {13}{2}}}{1664}+\frac {189 \left (3+2 x \right )^{\frac {15}{2}}}{640}-\frac {27 \left (3+2 x \right )^{\frac {17}{2}}}{2176}\) | \(74\) |
meijerg | \(-\frac {30 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+\frac {4 x}{3}\right ) \sqrt {1+\frac {2 x}{3}}}{3}\right )}{\sqrt {\pi }}-\frac {657 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (-2 x +2\right )}{15}\right )}{2 \sqrt {\pi }}-\frac {11745 \sqrt {3}\, \left (\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (\frac {20}{3} x^{2}-8 x +8\right )}{105}\right )}{8 \sqrt {\pi }}-\frac {108459 \sqrt {3}\, \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (-\frac {280}{27} x^{3}+\frac {40}{3} x^{2}-16 x +16\right )}{315}\right )}{32 \sqrt {\pi }}-\frac {132435 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{3465}-\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (\frac {560}{9} x^{4}-\frac {2240}{27} x^{3}+\frac {320}{3} x^{2}-128 x +128\right )}{3465}\right )}{32 \sqrt {\pi }}-\frac {72171 \sqrt {3}\, \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (-\frac {2464}{27} x^{5}+\frac {1120}{9} x^{4}-\frac {4480}{27} x^{3}+\frac {640}{3} x^{2}-256 x +256\right )}{9009}\right )}{32 \sqrt {\pi }}+\frac {177147 \sqrt {3}\, \left (-\frac {8192 \sqrt {\pi }}{109395}+\frac {4 \sqrt {\pi }\, \left (1+\frac {2 x}{3}\right )^{\frac {3}{2}} \left (-\frac {91520}{243} x^{7}+\frac {128128}{243} x^{6}-\frac {19712}{27} x^{5}+\frac {8960}{9} x^{4}-\frac {35840}{27} x^{3}+\frac {5120}{3} x^{2}-2048 x +2048\right )}{109395}\right )}{512 \sqrt {\pi }}\) | \(299\) |
-1/765765*(1216215*x^7-1702701*x^6-20968794*x^5-47286540*x^4-50880095*x^3- 29756385*x^2-9013014*x-1197186)*(3+2*x)^(3/2)
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.47 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {1}{765765} \, {\left (2432430 \, x^{8} + 243243 \, x^{7} - 47045691 \, x^{6} - 157479462 \, x^{5} - 243619810 \, x^{4} - 212153055 \, x^{3} - 107295183 \, x^{2} - 29433414 \, x - 3591558\right )} \sqrt {2 \, x + 3} \]
-1/765765*(2432430*x^8 + 243243*x^7 - 47045691*x^6 - 157479462*x^5 - 24361 9810*x^4 - 212153055*x^3 - 107295183*x^2 - 29433414*x - 3591558)*sqrt(2*x + 3)
Time = 1.01 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=- \frac {27 \left (2 x + 3\right )^{\frac {17}{2}}}{2176} + \frac {189 \left (2 x + 3\right )^{\frac {15}{2}}}{640} - \frac {3519 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {10475 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} - \frac {17201 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} + \frac {16005 \left (2 x + 3\right )^{\frac {7}{2}}}{896} - \frac {1585 \left (2 x + 3\right )^{\frac {5}{2}}}{128} + \frac {1625 \left (2 x + 3\right )^{\frac {3}{2}}}{384} \]
-27*(2*x + 3)**(17/2)/2176 + 189*(2*x + 3)**(15/2)/640 - 3519*(2*x + 3)**( 13/2)/1664 + 10475*(2*x + 3)**(11/2)/1408 - 17201*(2*x + 3)**(9/2)/1152 + 16005*(2*x + 3)**(7/2)/896 - 1585*(2*x + 3)**(5/2)/128 + 1625*(2*x + 3)**( 3/2)/384
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {27}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {189}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {3519}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {10475}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {17201}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {16005}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {1585}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {1625}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} \]
-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3) ^(13/2) + 10475/1408*(2*x + 3)^(11/2) - 17201/1152*(2*x + 3)^(9/2) + 16005 /896*(2*x + 3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/2)
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {27}{2176} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {189}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {3519}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {10475}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {17201}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {16005}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {1585}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {1625}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} \]
-27/2176*(2*x + 3)^(17/2) + 189/640*(2*x + 3)^(15/2) - 3519/1664*(2*x + 3) ^(13/2) + 10475/1408*(2*x + 3)^(11/2) - 17201/1152*(2*x + 3)^(9/2) + 16005 /896*(2*x + 3)^(7/2) - 1585/128*(2*x + 3)^(5/2) + 1625/384*(2*x + 3)^(3/2)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625\,{\left (2\,x+3\right )}^{3/2}}{384}-\frac {1585\,{\left (2\,x+3\right )}^{5/2}}{128}+\frac {16005\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {17201\,{\left (2\,x+3\right )}^{9/2}}{1152}+\frac {10475\,{\left (2\,x+3\right )}^{11/2}}{1408}-\frac {3519\,{\left (2\,x+3\right )}^{13/2}}{1664}+\frac {189\,{\left (2\,x+3\right )}^{15/2}}{640}-\frac {27\,{\left (2\,x+3\right )}^{17/2}}{2176} \]