Integrand size = 24, antiderivative size = 79 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {41503}{96} (1-2 x)^{3/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {675}{416} (1-2 x)^{13/2} \]
-41503/96*(1-2*x)^(3/2)+91091/160*(1-2*x)^(5/2)-5711/16*(1-2*x)^(7/2)+1949 /16*(1-2*x)^(9/2)-7695/352*(1-2*x)^(11/2)+675/416*(1-2*x)^(13/2)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=-\frac {(1-2 x)^{3/2} \left (253898+607254 x+913245 x^2+868215 x^3+471825 x^4+111375 x^5\right )}{2145} \]
-1/2145*((1 - 2*x)^(3/2)*(253898 + 607254*x + 913245*x^2 + 868215*x^3 + 47 1825*x^4 + 111375*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.
\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2 \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {675}{32} (1-2 x)^{11/2}+\frac {7695}{32} (1-2 x)^{9/2}-\frac {17541}{16} (1-2 x)^{7/2}+\frac {39977}{16} (1-2 x)^{5/2}-\frac {91091}{32} (1-2 x)^{3/2}+\frac {41503}{32} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {675}{416} (1-2 x)^{13/2}-\frac {7695}{352} (1-2 x)^{11/2}+\frac {1949}{16} (1-2 x)^{9/2}-\frac {5711}{16} (1-2 x)^{7/2}+\frac {91091}{160} (1-2 x)^{5/2}-\frac {41503}{96} (1-2 x)^{3/2}\) |
(-41503*(1 - 2*x)^(3/2))/96 + (91091*(1 - 2*x)^(5/2))/160 - (5711*(1 - 2*x )^(7/2))/16 + (1949*(1 - 2*x)^(9/2))/16 - (7695*(1 - 2*x)^(11/2))/352 + (6 75*(1 - 2*x)^(13/2))/416
3.19.4.3.1 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 = 0.96 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (111375 x^{5}+471825 x^{4}+868215 x^{3}+913245 x^{2}+607254 x +253898\right )}{2145}\) | \(35\) |
trager | \(\left (\frac {1350}{13} x^{6}+\frac {55485}{143} x^{5}+\frac {84307}{143} x^{4}+\frac {63885}{143} x^{3}+\frac {100421}{715} x^{2}-\frac {99458}{2145} x -\frac {253898}{2145}\right ) \sqrt {1-2 x}\) | \(39\) |
pseudoelliptic | \(\frac {\sqrt {1-2 x}\, \left (222750 x^{6}+832275 x^{5}+1264605 x^{4}+958275 x^{3}+301263 x^{2}-99458 x -253898\right )}{2145}\) | \(40\) |
risch | \(-\frac {\left (222750 x^{6}+832275 x^{5}+1264605 x^{4}+958275 x^{3}+301263 x^{2}-99458 x -253898\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\) | \(45\) |
derivativedivides | \(-\frac {41503 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {91091 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {5711 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1949 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {7695 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {675 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
default | \(-\frac {41503 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {91091 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {5711 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1949 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {7695 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {675 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
meijerg | \(\frac {24 \sqrt {\pi }-12 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {141 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {3532 \sqrt {\pi }}{105}-\frac {883 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{210}}{\sqrt {\pi }}-\frac {2763 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {384 \sqrt {\pi }}{77}-\frac {3 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{77}}{\sqrt {\pi }}-\frac {675 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}\) | \(220\) |
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=\frac {1}{2145} \, {\left (222750 \, x^{6} + 832275 \, x^{5} + 1264605 \, x^{4} + 958275 \, x^{3} + 301263 \, x^{2} - 99458 \, x - 253898\right )} \sqrt {-2 \, x + 1} \]
1/2145*(222750*x^6 + 832275*x^5 + 1264605*x^4 + 958275*x^3 + 301263*x^2 - 99458*x - 253898)*sqrt(-2*x + 1)
Time = 0.70 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=\frac {675 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {7695 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {1949 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {5711 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} + \frac {91091 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {41503 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \]
675*(1 - 2*x)**(13/2)/416 - 7695*(1 - 2*x)**(11/2)/352 + 1949*(1 - 2*x)**( 9/2)/16 - 5711*(1 - 2*x)**(7/2)/16 + 91091*(1 - 2*x)**(5/2)/160 - 41503*(1 - 2*x)**(3/2)/96
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=\frac {675}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {7695}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1949}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {91091}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {41503}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
675/416*(-2*x + 1)^(13/2) - 7695/352*(-2*x + 1)^(11/2) + 1949/16*(-2*x + 1 )^(9/2) - 5711/16*(-2*x + 1)^(7/2) + 91091/160*(-2*x + 1)^(5/2) - 41503/96 *(-2*x + 1)^(3/2)
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=\frac {675}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {7695}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1949}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {91091}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {41503}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
675/416*(2*x - 1)^6*sqrt(-2*x + 1) + 7695/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1949/16*(2*x - 1)^4*sqrt(-2*x + 1) + 5711/16*(2*x - 1)^3*sqrt(-2*x + 1) + 91091/160*(2*x - 1)^2*sqrt(-2*x + 1) - 41503/96*(-2*x + 1)^(3/2)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx=\frac {91091\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {41503\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {5711\,{\left (1-2\,x\right )}^{7/2}}{16}+\frac {1949\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {7695\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {675\,{\left (1-2\,x\right )}^{13/2}}{416} \]