Integrand size = 24, antiderivative size = 92 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {456533}{192} (1-2 x)^{3/2}+\frac {302379}{80} (1-2 x)^{5/2}-\frac {190707}{64} (1-2 x)^{7/2}+\frac {98209}{72} (1-2 x)^{9/2}-\frac {260055}{704} (1-2 x)^{11/2}+\frac {11475}{208} (1-2 x)^{13/2}-\frac {225}{64} (1-2 x)^{15/2} \]
-456533/192*(1-2*x)^(3/2)+302379/80*(1-2*x)^(5/2)-190707/64*(1-2*x)^(7/2)+ 98209/72*(1-2*x)^(9/2)-260055/704*(1-2*x)^(11/2)+11475/208*(1-2*x)^(13/2)- 225/64*(1-2*x)^(15/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {(1-2 x)^{3/2} \left (3420622+8871906 x+15577455 x^2+18934285 x^3+15061950 x^4+7016625 x^5+1447875 x^6\right )}{6435} \]
-1/6435*((1 - 2*x)^(3/2)*(3420622 + 8871906*x + 15577455*x^2 + 18934285*x^ 3 + 15061950*x^4 + 7016625*x^5 + 1447875*x^6))
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 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.
\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3 \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {3375}{64} (1-2 x)^{13/2}-\frac {11475}{16} (1-2 x)^{11/2}+\frac {260055}{64} (1-2 x)^{9/2}-\frac {98209}{8} (1-2 x)^{7/2}+\frac {1334949}{64} (1-2 x)^{5/2}-\frac {302379}{16} (1-2 x)^{3/2}+\frac {456533}{64} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {225}{64} (1-2 x)^{15/2}+\frac {11475}{208} (1-2 x)^{13/2}-\frac {260055}{704} (1-2 x)^{11/2}+\frac {98209}{72} (1-2 x)^{9/2}-\frac {190707}{64} (1-2 x)^{7/2}+\frac {302379}{80} (1-2 x)^{5/2}-\frac {456533}{192} (1-2 x)^{3/2}\) |
(-456533*(1 - 2*x)^(3/2))/192 + (302379*(1 - 2*x)^(5/2))/80 - (190707*(1 - 2*x)^(7/2))/64 + (98209*(1 - 2*x)^(9/2))/72 - (260055*(1 - 2*x)^(11/2))/7 04 + (11475*(1 - 2*x)^(13/2))/208 - (225*(1 - 2*x)^(15/2))/64
3.19.16.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.95 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (1447875 x^{6}+7016625 x^{5}+15061950 x^{4}+18934285 x^{3}+15577455 x^{2}+8871906 x +3420622\right )}{6435}\) | \(40\) |
trager | \(\left (450 x^{7}+\frac {25425}{13} x^{6}+\frac {513495}{143} x^{5}+\frac {4561324}{1287} x^{4}+\frac {2444125}{1287} x^{3}+\frac {722119}{2145} x^{2}-\frac {2030662}{6435} x -\frac {3420622}{6435}\right ) \sqrt {1-2 x}\) | \(44\) |
pseudoelliptic | \(\frac {\left (2895750 x^{7}+12585375 x^{6}+23107275 x^{5}+22806620 x^{4}+12220625 x^{3}+2166357 x^{2}-2030662 x -3420622\right ) \sqrt {1-2 x}}{6435}\) | \(45\) |
risch | \(-\frac {\left (2895750 x^{7}+12585375 x^{6}+23107275 x^{5}+22806620 x^{4}+12220625 x^{3}+2166357 x^{2}-2030662 x -3420622\right ) \left (-1+2 x \right )}{6435 \sqrt {1-2 x}}\) | \(50\) |
derivativedivides | \(-\frac {456533 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {302379 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {190707 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {260055 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {11475 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {225 \left (1-2 x \right )^{\frac {15}{2}}}{64}\) | \(65\) |
default | \(-\frac {456533 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {302379 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {190707 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {260055 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {11475 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {225 \left (1-2 x \right )^{\frac {15}{2}}}{64}\) | \(65\) |
meijerg | \(\frac {72 \sqrt {\pi }-36 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {513 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {5412 \sqrt {\pi }}{35}-\frac {1353 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{70}}{\sqrt {\pi }}-\frac {17119 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {328 \sqrt {\pi }}{7}-\frac {41 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{112}}{\sqrt {\pi }}-\frac {12825 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}+\frac {\frac {1200 \sqrt {\pi }}{1001}-\frac {75 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{64064}}{\sqrt {\pi }}\) | \(273\) |
-1/6435*(1-2*x)^(3/2)*(1447875*x^6+7016625*x^5+15061950*x^4+18934285*x^3+1 5577455*x^2+8871906*x+3420622)
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.48 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=\frac {1}{6435} \, {\left (2895750 \, x^{7} + 12585375 \, x^{6} + 23107275 \, x^{5} + 22806620 \, x^{4} + 12220625 \, x^{3} + 2166357 \, x^{2} - 2030662 \, x - 3420622\right )} \sqrt {-2 \, x + 1} \]
1/6435*(2895750*x^7 + 12585375*x^6 + 23107275*x^5 + 22806620*x^4 + 1222062 5*x^3 + 2166357*x^2 - 2030662*x - 3420622)*sqrt(-2*x + 1)
Time = 0.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=- \frac {225 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {11475 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {260055 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {98209 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {190707 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {302379 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {456533 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} \]
-225*(1 - 2*x)**(15/2)/64 + 11475*(1 - 2*x)**(13/2)/208 - 260055*(1 - 2*x) **(11/2)/704 + 98209*(1 - 2*x)**(9/2)/72 - 190707*(1 - 2*x)**(7/2)/64 + 30 2379*(1 - 2*x)**(5/2)/80 - 456533*(1 - 2*x)**(3/2)/192
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=-\frac {225}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {11475}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {260055}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {98209}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {190707}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {302379}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {456533}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
-225/64*(-2*x + 1)^(15/2) + 11475/208*(-2*x + 1)^(13/2) - 260055/704*(-2*x + 1)^(11/2) + 98209/72*(-2*x + 1)^(9/2) - 190707/64*(-2*x + 1)^(7/2) + 30 2379/80*(-2*x + 1)^(5/2) - 456533/192*(-2*x + 1)^(3/2)
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=\frac {225}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {11475}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {260055}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {98209}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {190707}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {302379}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {456533}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
225/64*(2*x - 1)^7*sqrt(-2*x + 1) + 11475/208*(2*x - 1)^6*sqrt(-2*x + 1) + 260055/704*(2*x - 1)^5*sqrt(-2*x + 1) + 98209/72*(2*x - 1)^4*sqrt(-2*x + 1) + 190707/64*(2*x - 1)^3*sqrt(-2*x + 1) + 302379/80*(2*x - 1)^2*sqrt(-2* x + 1) - 456533/192*(-2*x + 1)^(3/2)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3 \, dx=\frac {302379\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {456533\,{\left (1-2\,x\right )}^{3/2}}{192}-\frac {190707\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {260055\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {11475\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {225\,{\left (1-2\,x\right )}^{15/2}}{64} \]