Integrand size = 27, antiderivative size = 79 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {325}{288} (3+2 x)^{9/2}-\frac {1065}{352} (3+2 x)^{11/2}+\frac {651}{208} (3+2 x)^{13/2}-\frac {359}{240} (3+2 x)^{15/2}+\frac {165}{544} (3+2 x)^{17/2}-\frac {9}{608} (3+2 x)^{19/2} \]
325/288*(3+2*x)^(9/2)-1065/352*(3+2*x)^(11/2)+651/208*(3+2*x)^(13/2)-359/2 40*(3+2*x)^(15/2)+165/544*(3+2*x)^(17/2)-9/608*(3+2*x)^(19/2)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {(3+2 x)^{9/2} \left (-1670104-8846388 x-17037702 x^2-13495911 x^3-2702700 x^4+984555 x^5\right )}{2078505} \]
-1/2078505*((3 + 2*x)^(9/2)*(-1670104 - 8846388*x - 17037702*x^2 - 1349591 1*x^3 - 2702700*x^4 + 984555*x^5))
Time = 0.20 (sec) , antiderivative size = 79, 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.
\(\displaystyle \int (5-x) (2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {9}{32} (2 x+3)^{17/2}+\frac {165}{32} (2 x+3)^{15/2}-\frac {359}{16} (2 x+3)^{13/2}+\frac {651}{16} (2 x+3)^{11/2}-\frac {1065}{32} (2 x+3)^{9/2}+\frac {325}{32} (2 x+3)^{7/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9}{608} (2 x+3)^{19/2}+\frac {165}{544} (2 x+3)^{17/2}-\frac {359}{240} (2 x+3)^{15/2}+\frac {651}{208} (2 x+3)^{13/2}-\frac {1065}{352} (2 x+3)^{11/2}+\frac {325}{288} (2 x+3)^{9/2}\) |
(325*(3 + 2*x)^(9/2))/288 - (1065*(3 + 2*x)^(11/2))/352 + (651*(3 + 2*x)^( 13/2))/208 - (359*(3 + 2*x)^(15/2))/240 + (165*(3 + 2*x)^(17/2))/544 - (9* (3 + 2*x)^(19/2))/608
3.26.34.3.1 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 = 0.37 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (3+2 x \right )^{\frac {9}{2}}}{2078505}\) | \(35\) |
pseudoelliptic | \(-\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (3+2 x \right )^{\frac {9}{2}}}{2078505}\) | \(35\) |
trager | \(\left (-\frac {144}{19} x^{9}-\frac {7968}{323} x^{8}+\frac {612424}{4845} x^{7}+\frac {19589192}{20995} x^{6}+\frac {577368069}{230945} x^{5}+\frac {1538030804}{415701} x^{4}+\frac {456297481}{138567} x^{3}+\frac {405735126}{230945} x^{2}+\frac {119699988}{230945} x +\frac {15030936}{230945}\right ) \sqrt {3+2 x}\) | \(54\) |
risch | \(-\frac {\left (15752880 x^{9}+51274080 x^{8}-262729896 x^{7}-1939330008 x^{6}-5196312621 x^{5}-7690154020 x^{4}-6844462215 x^{3}-3651616134 x^{2}-1077299892 x -135278424\right ) \sqrt {3+2 x}}{2078505}\) | \(55\) |
derivativedivides | \(\frac {325 \left (3+2 x \right )^{\frac {9}{2}}}{288}-\frac {1065 \left (3+2 x \right )^{\frac {11}{2}}}{352}+\frac {651 \left (3+2 x \right )^{\frac {13}{2}}}{208}-\frac {359 \left (3+2 x \right )^{\frac {15}{2}}}{240}+\frac {165 \left (3+2 x \right )^{\frac {17}{2}}}{544}-\frac {9 \left (3+2 x \right )^{\frac {19}{2}}}{608}\) | \(56\) |
default | \(\frac {325 \left (3+2 x \right )^{\frac {9}{2}}}{288}-\frac {1065 \left (3+2 x \right )^{\frac {11}{2}}}{352}+\frac {651 \left (3+2 x \right )^{\frac {13}{2}}}{208}-\frac {359 \left (3+2 x \right )^{\frac {15}{2}}}{240}+\frac {165 \left (3+2 x \right )^{\frac {17}{2}}}{544}-\frac {9 \left (3+2 x \right )^{\frac {19}{2}}}{608}\) | \(56\) |
meijerg | \(\frac {12629925 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{135135}+\frac {16 \sqrt {\pi }\, \left (\frac {1408}{81} x^{6}+\frac {2560}{27} x^{5}+\frac {14656}{81} x^{4}+\frac {3392}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{135135}\right )}{128 \sqrt {\pi }}+\frac {25948755 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{675675}-\frac {4 \sqrt {\pi }\, \left (-\frac {146432}{729} x^{7}-\frac {259072}{243} x^{6}-\frac {52736}{27} x^{5}-\frac {102400}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{675675}\right )}{256 \sqrt {\pi }}+\frac {10333575 \sqrt {3}\, \left (-\frac {4096 \sqrt {\pi }}{11486475}+\frac {16 \sqrt {\pi }\, \left (\frac {366080}{729} x^{8}+\frac {1903616}{729} x^{7}+\frac {1129216}{243} x^{6}+\frac {25856}{9} x^{5}+\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{11486475}\right )}{512 \sqrt {\pi }}+\frac {76545 \sqrt {3}\, \left (\frac {64 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (-\frac {128}{27} x^{5}-\frac {2176}{81} x^{4}-\frac {1472}{27} x^{3}-\frac {128}{3} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{10395}\right )}{2 \sqrt {\pi }}+\frac {42525 \sqrt {3}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {16 \sqrt {\pi }\, \left (\frac {32}{81} x^{4}+\frac {64}{27} x^{3}+\frac {16}{3} x^{2}+\frac {16}{3} x +2\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{8 \sqrt {\pi }}-\frac {18600435 \sqrt {3}\, \left (\frac {8192 \sqrt {\pi }}{43648605}-\frac {8 \sqrt {\pi }\, \left (-\frac {4978688}{2187} x^{9}-\frac {8493056}{729} x^{8}-\frac {14789632}{729} x^{7}-\frac {2951168}{243} x^{6}-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{43648605}\right )}{1024 \sqrt {\pi }}\) | \(353\) |
-1/2078505*(984555*x^5-2702700*x^4-13495911*x^3-17037702*x^2-8846388*x-167 0104)*(3+2*x)^(9/2)
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {1}{2078505} \, {\left (15752880 \, x^{9} + 51274080 \, x^{8} - 262729896 \, x^{7} - 1939330008 \, x^{6} - 5196312621 \, x^{5} - 7690154020 \, x^{4} - 6844462215 \, x^{3} - 3651616134 \, x^{2} - 1077299892 \, x - 135278424\right )} \sqrt {2 \, x + 3} \]
-1/2078505*(15752880*x^9 + 51274080*x^8 - 262729896*x^7 - 1939330008*x^6 - 5196312621*x^5 - 7690154020*x^4 - 6844462215*x^3 - 3651616134*x^2 - 10772 99892*x - 135278424)*sqrt(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (70) = 140\).
Time = 0.47 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.85 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=- \frac {144 x^{9} \sqrt {2 x + 3}}{19} - \frac {7968 x^{8} \sqrt {2 x + 3}}{323} + \frac {612424 x^{7} \sqrt {2 x + 3}}{4845} + \frac {19589192 x^{6} \sqrt {2 x + 3}}{20995} + \frac {577368069 x^{5} \sqrt {2 x + 3}}{230945} + \frac {1538030804 x^{4} \sqrt {2 x + 3}}{415701} + \frac {456297481 x^{3} \sqrt {2 x + 3}}{138567} + \frac {405735126 x^{2} \sqrt {2 x + 3}}{230945} + \frac {119699988 x \sqrt {2 x + 3}}{230945} + \frac {15030936 \sqrt {2 x + 3}}{230945} \]
-144*x**9*sqrt(2*x + 3)/19 - 7968*x**8*sqrt(2*x + 3)/323 + 612424*x**7*sqr t(2*x + 3)/4845 + 19589192*x**6*sqrt(2*x + 3)/20995 + 577368069*x**5*sqrt( 2*x + 3)/230945 + 1538030804*x**4*sqrt(2*x + 3)/415701 + 456297481*x**3*sq rt(2*x + 3)/138567 + 405735126*x**2*sqrt(2*x + 3)/230945 + 119699988*x*sqr t(2*x + 3)/230945 + 15030936*sqrt(2*x + 3)/230945
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]
-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15 /2) + 651/208*(2*x + 3)^(13/2) - 1065/352*(2*x + 3)^(11/2) + 325/288*(2*x + 3)^(9/2)
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]
-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15 /2) + 651/208*(2*x + 3)^(13/2) - 1065/352*(2*x + 3)^(11/2) + 325/288*(2*x + 3)^(9/2)
Time = 11.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {325\,{\left (2\,x+3\right )}^{9/2}}{288}-\frac {1065\,{\left (2\,x+3\right )}^{11/2}}{352}+\frac {651\,{\left (2\,x+3\right )}^{13/2}}{208}-\frac {359\,{\left (2\,x+3\right )}^{15/2}}{240}+\frac {165\,{\left (2\,x+3\right )}^{17/2}}{544}-\frac {9\,{\left (2\,x+3\right )}^{19/2}}{608} \]