Integrand size = 27, antiderivative size = 105 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=\frac {1625}{128} \sqrt {3+2 x}-\frac {7925}{384} (3+2 x)^{3/2}+\frac {3201}{128} (3+2 x)^{5/2}-\frac {17201}{896} (3+2 x)^{7/2}+\frac {10475 (3+2 x)^{9/2}}{1152}-\frac {3519 (3+2 x)^{11/2}}{1408}+\frac {567 (3+2 x)^{13/2}}{1664}-\frac {9}{640} (3+2 x)^{15/2} \]
-7925/384*(3+2*x)^(3/2)+3201/128*(3+2*x)^(5/2)-17201/896*(3+2*x)^(7/2)+104 75/1152*(3+2*x)^(9/2)-3519/1408*(3+2*x)^(11/2)+567/1664*(3+2*x)^(13/2)-9/6 40*(3+2*x)^(15/2)+1625/128*(3+2*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=-\frac {\sqrt {3+2 x} \left (-196506-535098 x-1924641 x^2-3285105 x^3-3109960 x^4-1407294 x^5-130977 x^6+81081 x^7\right )}{45045} \]
-1/45045*(Sqrt[3 + 2*x]*(-196506 - 535098*x - 1924641*x^2 - 3285105*x^3 - 3109960*x^4 - 1407294*x^5 - 130977*x^6 + 81081*x^7))
Time = 0.21 (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.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^3}{\sqrt {2 x+3}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{13/2}+\frac {567}{128} (2 x+3)^{11/2}-\frac {3519}{128} (2 x+3)^{9/2}+\frac {10475}{128} (2 x+3)^{7/2}-\frac {17201}{128} (2 x+3)^{5/2}+\frac {16005}{128} (2 x+3)^{3/2}-\frac {7925}{128} \sqrt {2 x+3}+\frac {1625}{128 \sqrt {2 x+3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9}{640} (2 x+3)^{15/2}+\frac {567 (2 x+3)^{13/2}}{1664}-\frac {3519 (2 x+3)^{11/2}}{1408}+\frac {10475 (2 x+3)^{9/2}}{1152}-\frac {17201}{896} (2 x+3)^{7/2}+\frac {3201}{128} (2 x+3)^{5/2}-\frac {7925}{384} (2 x+3)^{3/2}+\frac {1625}{128} \sqrt {2 x+3}\) |
(1625*Sqrt[3 + 2*x])/128 - (7925*(3 + 2*x)^(3/2))/384 + (3201*(3 + 2*x)^(5 /2))/128 - (17201*(3 + 2*x)^(7/2))/896 + (10475*(3 + 2*x)^(9/2))/1152 - (3 519*(3 + 2*x)^(11/2))/1408 + (567*(3 + 2*x)^(13/2))/1664 - (9*(3 + 2*x)^(1 5/2))/640
3.26.46.3.1 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 = 44, normalized size of antiderivative = 0.42
method | result | size |
trager | \(\left (-\frac {9}{5} x^{7}+\frac {189}{65} x^{6}+\frac {22338}{715} x^{5}+\frac {88856}{1287} x^{4}+\frac {219007}{3003} x^{3}+\frac {213849}{5005} x^{2}+\frac {178366}{15015} x +\frac {21834}{5005}\right ) \sqrt {3+2 x}\) | \(44\) |
gosper | \(-\frac {\left (81081 x^{7}-130977 x^{6}-1407294 x^{5}-3109960 x^{4}-3285105 x^{3}-1924641 x^{2}-535098 x -196506\right ) \sqrt {3+2 x}}{45045}\) | \(45\) |
risch | \(-\frac {\left (81081 x^{7}-130977 x^{6}-1407294 x^{5}-3109960 x^{4}-3285105 x^{3}-1924641 x^{2}-535098 x -196506\right ) \sqrt {3+2 x}}{45045}\) | \(45\) |
pseudoelliptic | \(-\frac {\left (81081 x^{7}-130977 x^{6}-1407294 x^{5}-3109960 x^{4}-3285105 x^{3}-1924641 x^{2}-535098 x -196506\right ) \sqrt {3+2 x}}{45045}\) | \(45\) |
derivativedivides | \(-\frac {7925 \left (3+2 x \right )^{\frac {3}{2}}}{384}+\frac {3201 \left (3+2 x \right )^{\frac {5}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {7}{2}}}{896}+\frac {10475 \left (3+2 x \right )^{\frac {9}{2}}}{1152}-\frac {3519 \left (3+2 x \right )^{\frac {11}{2}}}{1408}+\frac {567 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {9 \left (3+2 x \right )^{\frac {15}{2}}}{640}+\frac {1625 \sqrt {3+2 x}}{128}\) | \(74\) |
default | \(-\frac {7925 \left (3+2 x \right )^{\frac {3}{2}}}{384}+\frac {3201 \left (3+2 x \right )^{\frac {5}{2}}}{128}-\frac {17201 \left (3+2 x \right )^{\frac {7}{2}}}{896}+\frac {10475 \left (3+2 x \right )^{\frac {9}{2}}}{1152}-\frac {3519 \left (3+2 x \right )^{\frac {11}{2}}}{1408}+\frac {567 \left (3+2 x \right )^{\frac {13}{2}}}{1664}-\frac {9 \left (3+2 x \right )^{\frac {15}{2}}}{640}+\frac {1625 \sqrt {3+2 x}}{128}\) | \(74\) |
meijerg | \(\frac {20 \sqrt {3}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {2 x}{3}}\right )}{\sqrt {\pi }}+\frac {219 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8 x}{3}+8\right ) \sqrt {1+\frac {2 x}{3}}}{6}\right )}{\sqrt {\pi }}+\frac {3915 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{4 \sqrt {\pi }}+\frac {36153 \sqrt {3}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{140}\right )}{16 \sqrt {\pi }}+\frac {44145 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{315}\right )}{16 \sqrt {\pi }}+\frac {24057 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{693}-\frac {\sqrt {\pi }\, \left (-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{1386}\right )}{16 \sqrt {\pi }}-\frac {59049 \sqrt {3}\, \left (\frac {4096 \sqrt {\pi }}{6435}-\frac {\sqrt {\pi }\, \left (-\frac {292864}{729} x^{7}+\frac {157696}{243} x^{6}-\frac {28672}{27} x^{5}+\frac {143360}{81} x^{4}-\frac {81920}{27} x^{3}+\frac {16384}{3} x^{2}-\frac {32768}{3} x +32768\right ) \sqrt {1+\frac {2 x}{3}}}{51480}\right )}{256 \sqrt {\pi }}\) | \(294\) |
(-9/5*x^7+189/65*x^6+22338/715*x^5+88856/1287*x^4+219007/3003*x^3+213849/5 005*x^2+178366/15015*x+21834/5005)*(3+2*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=-\frac {1}{45045} \, {\left (81081 \, x^{7} - 130977 \, x^{6} - 1407294 \, x^{5} - 3109960 \, x^{4} - 3285105 \, x^{3} - 1924641 \, x^{2} - 535098 \, x - 196506\right )} \sqrt {2 \, x + 3} \]
-1/45045*(81081*x^7 - 130977*x^6 - 1407294*x^5 - 3109960*x^4 - 3285105*x^3 - 1924641*x^2 - 535098*x - 196506)*sqrt(2*x + 3)
Time = 1.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=- \frac {9 \left (2 x + 3\right )^{\frac {15}{2}}}{640} + \frac {567 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} - \frac {3519 \left (2 x + 3\right )^{\frac {11}{2}}}{1408} + \frac {10475 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} - \frac {17201 \left (2 x + 3\right )^{\frac {7}{2}}}{896} + \frac {3201 \left (2 x + 3\right )^{\frac {5}{2}}}{128} - \frac {7925 \left (2 x + 3\right )^{\frac {3}{2}}}{384} + \frac {1625 \sqrt {2 x + 3}}{128} \]
-9*(2*x + 3)**(15/2)/640 + 567*(2*x + 3)**(13/2)/1664 - 3519*(2*x + 3)**(1 1/2)/1408 + 10475*(2*x + 3)**(9/2)/1152 - 17201*(2*x + 3)**(7/2)/896 + 320 1*(2*x + 3)**(5/2)/128 - 7925*(2*x + 3)**(3/2)/384 + 1625*sqrt(2*x + 3)/12 8
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=-\frac {9}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {567}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {3519}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {10475}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {17201}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {3201}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {7925}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {1625}{128} \, \sqrt {2 \, x + 3} \]
-9/640*(2*x + 3)^(15/2) + 567/1664*(2*x + 3)^(13/2) - 3519/1408*(2*x + 3)^ (11/2) + 10475/1152*(2*x + 3)^(9/2) - 17201/896*(2*x + 3)^(7/2) + 3201/128 *(2*x + 3)^(5/2) - 7925/384*(2*x + 3)^(3/2) + 1625/128*sqrt(2*x + 3)
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=-\frac {9}{640} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {567}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {3519}{1408} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {10475}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {17201}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {3201}{128} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {7925}{384} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {1625}{128} \, \sqrt {2 \, x + 3} \]
-9/640*(2*x + 3)^(15/2) + 567/1664*(2*x + 3)^(13/2) - 3519/1408*(2*x + 3)^ (11/2) + 10475/1152*(2*x + 3)^(9/2) - 17201/896*(2*x + 3)^(7/2) + 3201/128 *(2*x + 3)^(5/2) - 7925/384*(2*x + 3)^(3/2) + 1625/128*sqrt(2*x + 3)
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^3}{\sqrt {3+2 x}} \, dx=\frac {1625\,\sqrt {2\,x+3}}{128}-\frac {7925\,{\left (2\,x+3\right )}^{3/2}}{384}+\frac {3201\,{\left (2\,x+3\right )}^{5/2}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{7/2}}{896}+\frac {10475\,{\left (2\,x+3\right )}^{9/2}}{1152}-\frac {3519\,{\left (2\,x+3\right )}^{11/2}}{1408}+\frac {567\,{\left (2\,x+3\right )}^{13/2}}{1664}-\frac {9\,{\left (2\,x+3\right )}^{15/2}}{640} \]