Integrand size = 27, antiderivative size = 105 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=-\frac {1625}{128 \sqrt {3+2 x}}-\frac {7925}{128} \sqrt {3+2 x}+\frac {5335}{128} (3+2 x)^{3/2}-\frac {17201}{640} (3+2 x)^{5/2}+\frac {10475}{896} (3+2 x)^{7/2}-\frac {391}{128} (3+2 x)^{9/2}+\frac {567 (3+2 x)^{11/2}}{1408}-\frac {27 (3+2 x)^{13/2}}{1664} \]
5335/128*(3+2*x)^(3/2)-17201/640*(3+2*x)^(5/2)+10475/896*(3+2*x)^(7/2)-391 /128*(3+2*x)^(9/2)+567/1408*(3+2*x)^(11/2)-27/1664*(3+2*x)^(13/2)-1625/128 /(3+2*x)^(1/2)-7925/128*(3+2*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=-\frac {431614+77138 x-256433 x^2-398339 x^3-392500 x^4-180530 x^5-19845 x^6+10395 x^7}{5005 \sqrt {3+2 x}} \]
-1/5005*(431614 + 77138*x - 256433*x^2 - 398339*x^3 - 392500*x^4 - 180530* x^5 - 19845*x^6 + 10395*x^7)/Sqrt[3 + 2*x]
Time = 0.23 (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.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^3}{(2 x+3)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{11/2}+\frac {567}{128} (2 x+3)^{9/2}-\frac {3519}{128} (2 x+3)^{7/2}+\frac {10475}{128} (2 x+3)^{5/2}-\frac {17201}{128} (2 x+3)^{3/2}+\frac {16005}{128} \sqrt {2 x+3}-\frac {7925}{128 \sqrt {2 x+3}}+\frac {1625}{128 (2 x+3)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {27 (2 x+3)^{13/2}}{1664}+\frac {567 (2 x+3)^{11/2}}{1408}-\frac {391}{128} (2 x+3)^{9/2}+\frac {10475}{896} (2 x+3)^{7/2}-\frac {17201}{640} (2 x+3)^{5/2}+\frac {5335}{128} (2 x+3)^{3/2}-\frac {7925}{128} \sqrt {2 x+3}-\frac {1625}{128 \sqrt {2 x+3}}\) |
-1625/(128*Sqrt[3 + 2*x]) - (7925*Sqrt[3 + 2*x])/128 + (5335*(3 + 2*x)^(3/ 2))/128 - (17201*(3 + 2*x)^(5/2))/640 + (10475*(3 + 2*x)^(7/2))/896 - (391 *(3 + 2*x)^(9/2))/128 + (567*(3 + 2*x)^(11/2))/1408 - (27*(3 + 2*x)^(13/2) )/1664
3.26.47.3.1 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 = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {10395 x^{7}-19845 x^{6}-180530 x^{5}-392500 x^{4}-398339 x^{3}-256433 x^{2}+77138 x +431614}{5005 \sqrt {3+2 x}}\) | \(45\) |
trager | \(-\frac {10395 x^{7}-19845 x^{6}-180530 x^{5}-392500 x^{4}-398339 x^{3}-256433 x^{2}+77138 x +431614}{5005 \sqrt {3+2 x}}\) | \(45\) |
risch | \(-\frac {10395 x^{7}-19845 x^{6}-180530 x^{5}-392500 x^{4}-398339 x^{3}-256433 x^{2}+77138 x +431614}{5005 \sqrt {3+2 x}}\) | \(45\) |
pseudoelliptic | \(\frac {-10395 x^{7}+19845 x^{6}+180530 x^{5}+392500 x^{4}+398339 x^{3}+256433 x^{2}-77138 x -431614}{5005 \sqrt {3+2 x}}\) | \(45\) |
derivativedivides | \(\frac {5335 \left (3+2 x \right )^{\frac {3}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {5}{2}}}{640}+\frac {10475 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {391 \left (3+2 x \right )^{\frac {9}{2}}}{128}+\frac {567 \left (3+2 x \right )^{\frac {11}{2}}}{1408}-\frac {27 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {1625}{128 \sqrt {3+2 x}}-\frac {7925 \sqrt {3+2 x}}{128}\) | \(74\) |
default | \(\frac {5335 \left (3+2 x \right )^{\frac {3}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {5}{2}}}{640}+\frac {10475 \left (3+2 x \right )^{\frac {7}{2}}}{896}-\frac {391 \left (3+2 x \right )^{\frac {9}{2}}}{128}+\frac {567 \left (3+2 x \right )^{\frac {11}{2}}}{1408}-\frac {27 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {1625}{128 \sqrt {3+2 x}}-\frac {7925 \sqrt {3+2 x}}{128}\) | \(74\) |
meijerg | \(\frac {40 \sqrt {3}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {2 x}{3}}}\right )}{3 \sqrt {\pi }}+\frac {146 \sqrt {3}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {8 x}{3}+8\right )}{4 \sqrt {1+\frac {2 x}{3}}}\right )}{\sqrt {\pi }}+\frac {1305 \sqrt {3}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8}{9} x^{2}+\frac {16}{3} x +16\right )}{6 \sqrt {1+\frac {2 x}{3}}}\right )}{2 \sqrt {\pi }}+\frac {12051 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (\frac {64}{27} x^{3}-\frac {64}{9} x^{2}+\frac {128}{3} x +128\right )}{40 \sqrt {1+\frac {2 x}{3}}}\right )}{8 \sqrt {\pi }}+\frac {14715 \sqrt {3}\, \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {160}{81} x^{4}+\frac {128}{27} x^{3}-\frac {128}{9} x^{2}+\frac {256}{3} x +256\right )}{70 \sqrt {1+\frac {2 x}{3}}}\right )}{8 \sqrt {\pi }}+\frac {8019 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{63}+\frac {\sqrt {\pi }\, \left (\frac {896}{243} x^{5}-\frac {640}{81} x^{4}+\frac {512}{27} x^{3}-\frac {512}{9} x^{2}+\frac {1024}{3} x +1024\right )}{252 \sqrt {1+\frac {2 x}{3}}}\right )}{8 \sqrt {\pi }}-\frac {19683 \sqrt {3}\, \left (-\frac {2048 \sqrt {\pi }}{429}+\frac {\sqrt {\pi }\, \left (\frac {22528}{729} x^{7}-\frac {14336}{243} x^{6}+\frac {28672}{243} x^{5}-\frac {20480}{81} x^{4}+\frac {16384}{27} x^{3}-\frac {16384}{9} x^{2}+\frac {32768}{3} x +32768\right )}{6864 \sqrt {1+\frac {2 x}{3}}}\right )}{128 \sqrt {\pi }}\) | \(292\) |
-1/5005*(10395*x^7-19845*x^6-180530*x^5-392500*x^4-398339*x^3-256433*x^2+7 7138*x+431614)/(3+2*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=-\frac {10395 \, x^{7} - 19845 \, x^{6} - 180530 \, x^{5} - 392500 \, x^{4} - 398339 \, x^{3} - 256433 \, x^{2} + 77138 \, x + 431614}{5005 \, \sqrt {2 \, x + 3}} \]
-1/5005*(10395*x^7 - 19845*x^6 - 180530*x^5 - 392500*x^4 - 398339*x^3 - 25 6433*x^2 + 77138*x + 431614)/sqrt(2*x + 3)
Time = 1.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=- \frac {27 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {567 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} - \frac {391 \left (2 x + 3\right )^{\frac {9}{2}}}{128} + \frac {10475 \left (2 x + 3\right )^{\frac {7}{2}}}{896} - \frac {17201 \left (2 x + 3\right )^{\frac {5}{2}}}{640} + \frac {5335 \left (2 x + 3\right )^{\frac {3}{2}}}{128} - \frac {7925 \sqrt {2 x + 3}}{128} - \frac {1625}{128 \sqrt {2 x + 3}} \]
-27*(2*x + 3)**(13/2)/1664 + 567*(2*x + 3)**(11/2)/1408 - 391*(2*x + 3)**( 9/2)/128 + 10475*(2*x + 3)**(7/2)/896 - 17201*(2*x + 3)**(5/2)/640 + 5335* (2*x + 3)**(3/2)/128 - 7925*sqrt(2*x + 3)/128 - 1625/(128*sqrt(2*x + 3))
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=-\frac {27}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {567}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {391}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {10475}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {17201}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {5335}{128} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {7925}{128} \, \sqrt {2 \, x + 3} - \frac {1625}{128 \, \sqrt {2 \, x + 3}} \]
-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^ (9/2) + 10475/896*(2*x + 3)^(7/2) - 17201/640*(2*x + 3)^(5/2) + 5335/128*( 2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=-\frac {27}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {567}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {391}{128} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {10475}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - \frac {17201}{640} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {5335}{128} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {7925}{128} \, \sqrt {2 \, x + 3} - \frac {1625}{128 \, \sqrt {2 \, x + 3}} \]
-27/1664*(2*x + 3)^(13/2) + 567/1408*(2*x + 3)^(11/2) - 391/128*(2*x + 3)^ (9/2) + 10475/896*(2*x + 3)^(7/2) - 17201/640*(2*x + 3)^(5/2) + 5335/128*( 2*x + 3)^(3/2) - 7925/128*sqrt(2*x + 3) - 1625/128/sqrt(2*x + 3)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{(3+2 x)^{3/2}} \, dx=\frac {5335\,{\left (2\,x+3\right )}^{3/2}}{128}-\frac {7925\,\sqrt {2\,x+3}}{128}-\frac {1625}{128\,\sqrt {2\,x+3}}-\frac {17201\,{\left (2\,x+3\right )}^{5/2}}{640}+\frac {10475\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {391\,{\left (2\,x+3\right )}^{9/2}}{128}+\frac {567\,{\left (2\,x+3\right )}^{11/2}}{1408}-\frac {27\,{\left (2\,x+3\right )}^{13/2}}{1664} \]