Integrand size = 24, antiderivative size = 79 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=-\frac {65219}{96} (1-2 x)^{3/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {1125}{416} (1-2 x)^{13/2} \]
-65219/96*(1-2*x)^(3/2)+144837/160*(1-2*x)^(5/2)-64317/112*(1-2*x)^(7/2)+2 8555/144*(1-2*x)^(9/2)-12675/352*(1-2*x)^(11/2)+1125/416*(1-2*x)^(13/2)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=-\frac {(1-2 x)^{3/2} \left (8261156+19918608 x+30337080 x^2+29300075 x^3+16206750 x^4+3898125 x^5\right )}{45045} \]
-1/45045*((1 - 2*x)^(3/2)*(8261156 + 19918608*x + 30337080*x^2 + 29300075* x^3 + 16206750*x^4 + 3898125*x^5))
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3 \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {1125}{32} (1-2 x)^{11/2}+\frac {12675}{32} (1-2 x)^{9/2}-\frac {28555}{16} (1-2 x)^{7/2}+\frac {64317}{16} (1-2 x)^{5/2}-\frac {144837}{32} (1-2 x)^{3/2}+\frac {65219}{32} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1125}{416} (1-2 x)^{13/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {65219}{96} (1-2 x)^{3/2}\) |
(-65219*(1 - 2*x)^(3/2))/96 + (144837*(1 - 2*x)^(5/2))/160 - (64317*(1 - 2 *x)^(7/2))/112 + (28555*(1 - 2*x)^(9/2))/144 - (12675*(1 - 2*x)^(11/2))/35 2 + (1125*(1 - 2*x)^(13/2))/416
3.19.17.3.1 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 = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3898125 x^{5}+16206750 x^{4}+29300075 x^{3}+30337080 x^{2}+19918608 x +8261156\right )}{45045}\) | \(35\) |
trager | \(\left (\frac {2250}{13} x^{6}+\frac {90525}{143} x^{5}+\frac {1211240}{1287} x^{4}+\frac {6274817}{9009} x^{3}+\frac {3166712}{15015} x^{2}-\frac {3396296}{45045} x -\frac {8261156}{45045}\right ) \sqrt {1-2 x}\) | \(39\) |
pseudoelliptic | \(\frac {\sqrt {1-2 x}\, \left (7796250 x^{6}+28515375 x^{5}+42393400 x^{4}+31374085 x^{3}+9500136 x^{2}-3396296 x -8261156\right )}{45045}\) | \(40\) |
risch | \(-\frac {\left (7796250 x^{6}+28515375 x^{5}+42393400 x^{4}+31374085 x^{3}+9500136 x^{2}-3396296 x -8261156\right ) \left (-1+2 x \right )}{45045 \sqrt {1-2 x}}\) | \(45\) |
derivativedivides | \(-\frac {65219 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {144837 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {64317 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {28555 \left (1-2 x \right )^{\frac {9}{2}}}{144}-\frac {12675 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {1125 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
default | \(-\frac {65219 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {144837 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {64317 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {28555 \left (1-2 x \right )^{\frac {9}{2}}}{144}-\frac {12675 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {1125 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
meijerg | \(\frac {36 \sqrt {\pi }-18 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {108 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {1842 \sqrt {\pi }}{35}-\frac {921 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{140}}{\sqrt {\pi }}-\frac {4415 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {1880 \sqrt {\pi }}{231}-\frac {235 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{3696}}{\sqrt {\pi }}-\frac {1125 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}\) | \(220\) |
-1/45045*(1-2*x)^(3/2)*(3898125*x^5+16206750*x^4+29300075*x^3+30337080*x^2 +19918608*x+8261156)
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=\frac {1}{45045} \, {\left (7796250 \, x^{6} + 28515375 \, x^{5} + 42393400 \, x^{4} + 31374085 \, x^{3} + 9500136 \, x^{2} - 3396296 \, x - 8261156\right )} \sqrt {-2 \, x + 1} \]
1/45045*(7796250*x^6 + 28515375*x^5 + 42393400*x^4 + 31374085*x^3 + 950013 6*x^2 - 3396296*x - 8261156)*sqrt(-2*x + 1)
Time = 0.69 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=\frac {1125 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {12675 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {28555 \left (1 - 2 x\right )^{\frac {9}{2}}}{144} - \frac {64317 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {144837 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {65219 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \]
1125*(1 - 2*x)**(13/2)/416 - 12675*(1 - 2*x)**(11/2)/352 + 28555*(1 - 2*x) **(9/2)/144 - 64317*(1 - 2*x)**(7/2)/112 + 144837*(1 - 2*x)**(5/2)/160 - 6 5219*(1 - 2*x)**(3/2)/96
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=\frac {1125}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {12675}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {28555}{144} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {64317}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {144837}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
1125/416*(-2*x + 1)^(13/2) - 12675/352*(-2*x + 1)^(11/2) + 28555/144*(-2*x + 1)^(9/2) - 64317/112*(-2*x + 1)^(7/2) + 144837/160*(-2*x + 1)^(5/2) - 6 5219/96*(-2*x + 1)^(3/2)
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=\frac {1125}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {12675}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {28555}{144} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {64317}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {144837}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
1125/416*(2*x - 1)^6*sqrt(-2*x + 1) + 12675/352*(2*x - 1)^5*sqrt(-2*x + 1) + 28555/144*(2*x - 1)^4*sqrt(-2*x + 1) + 64317/112*(2*x - 1)^3*sqrt(-2*x + 1) + 144837/160*(2*x - 1)^2*sqrt(-2*x + 1) - 65219/96*(-2*x + 1)^(3/2)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx=\frac {144837\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {65219\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {64317\,{\left (1-2\,x\right )}^{7/2}}{112}+\frac {28555\,{\left (1-2\,x\right )}^{9/2}}{144}-\frac {12675\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {1125\,{\left (1-2\,x\right )}^{13/2}}{416} \]