Integrand size = 22, antiderivative size = 92 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=-\frac {184877}{192} (1-2 x)^{3/2}+\frac {12005}{8} (1-2 x)^{5/2}-\frac {74235}{64} (1-2 x)^{7/2}+\frac {4165}{8} (1-2 x)^{9/2}-\frac {97335}{704} (1-2 x)^{11/2}+\frac {81}{4} (1-2 x)^{13/2}-\frac {81}{64} (1-2 x)^{15/2} \]
-184877/192*(1-2*x)^(3/2)+12005/8*(1-2*x)^(5/2)-74235/64*(1-2*x)^(7/2)+416 5/8*(1-2*x)^(9/2)-97335/704*(1-2*x)^(11/2)+81/4*(1-2*x)^(13/2)-81/64*(1-2* x)^(15/2)
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=-\frac {1}{33} (1-2 x)^{3/2} \left (7288+18696 x+32220 x^2+38220 x^3+29565 x^4+13365 x^5+2673 x^6\right ) \]
-1/33*((1 - 2*x)^(3/2)*(7288 + 18696*x + 32220*x^2 + 38220*x^3 + 29565*x^4 + 13365*x^5 + 2673*x^6))
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {86, 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.
\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^5 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {1215}{64} (1-2 x)^{13/2}-\frac {1053}{4} (1-2 x)^{11/2}+\frac {97335}{64} (1-2 x)^{9/2}-\frac {37485}{8} (1-2 x)^{7/2}+\frac {519645}{64} (1-2 x)^{5/2}-\frac {60025}{8} (1-2 x)^{3/2}+\frac {184877}{64} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {81}{64} (1-2 x)^{15/2}+\frac {81}{4} (1-2 x)^{13/2}-\frac {97335}{704} (1-2 x)^{11/2}+\frac {4165}{8} (1-2 x)^{9/2}-\frac {74235}{64} (1-2 x)^{7/2}+\frac {12005}{8} (1-2 x)^{5/2}-\frac {184877}{192} (1-2 x)^{3/2}\) |
(-184877*(1 - 2*x)^(3/2))/192 + (12005*(1 - 2*x)^(5/2))/8 - (74235*(1 - 2* x)^(7/2))/64 + (4165*(1 - 2*x)^(9/2))/8 - (97335*(1 - 2*x)^(11/2))/704 + ( 81*(1 - 2*x)^(13/2))/4 - (81*(1 - 2*x)^(15/2))/64
3.18.91.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.95 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (2673 x^{6}+13365 x^{5}+29565 x^{4}+38220 x^{3}+32220 x^{2}+18696 x +7288\right )}{33}\) | \(40\) |
trager | \(\left (162 x^{7}+729 x^{6}+\frac {15255}{11} x^{5}+\frac {15625}{11} x^{4}+\frac {8740}{11} x^{3}+\frac {1724}{11} x^{2}-\frac {4120}{33} x -\frac {7288}{33}\right ) \sqrt {1-2 x}\) | \(44\) |
pseudoelliptic | \(\frac {\left (5346 x^{7}+24057 x^{6}+45765 x^{5}+46875 x^{4}+26220 x^{3}+5172 x^{2}-4120 x -7288\right ) \sqrt {1-2 x}}{33}\) | \(45\) |
risch | \(-\frac {\left (5346 x^{7}+24057 x^{6}+45765 x^{5}+46875 x^{4}+26220 x^{3}+5172 x^{2}-4120 x -7288\right ) \left (-1+2 x \right )}{33 \sqrt {1-2 x}}\) | \(50\) |
derivativedivides | \(-\frac {184877 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {12005 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {74235 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {4165 \left (1-2 x \right )^{\frac {9}{2}}}{8}-\frac {97335 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{4}-\frac {81 \left (1-2 x \right )^{\frac {15}{2}}}{64}\) | \(65\) |
default | \(-\frac {184877 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {12005 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {74235 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {4165 \left (1-2 x \right )^{\frac {9}{2}}}{8}-\frac {97335 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{4}-\frac {81 \left (1-2 x \right )^{\frac {15}{2}}}{64}\) | \(65\) |
meijerg | \(\frac {32 \sqrt {\pi }-16 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {110 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {64 \sqrt {\pi }-8 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{\sqrt {\pi }}-\frac {855 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{4 \sqrt {\pi }}+\frac {\frac {1392 \sqrt {\pi }}{77}-\frac {87 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{616}}{\sqrt {\pi }}-\frac {4779 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}+\frac {\frac {432 \sqrt {\pi }}{1001}-\frac {27 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{64064}}{\sqrt {\pi }}\) | \(273\) |
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.48 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=\frac {1}{33} \, {\left (5346 \, x^{7} + 24057 \, x^{6} + 45765 \, x^{5} + 46875 \, x^{4} + 26220 \, x^{3} + 5172 \, x^{2} - 4120 \, x - 7288\right )} \sqrt {-2 \, x + 1} \]
1/33*(5346*x^7 + 24057*x^6 + 45765*x^5 + 46875*x^4 + 26220*x^3 + 5172*x^2 - 4120*x - 7288)*sqrt(-2*x + 1)
Time = 0.73 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=- \frac {81 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {81 \left (1 - 2 x\right )^{\frac {13}{2}}}{4} - \frac {97335 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {4165 \left (1 - 2 x\right )^{\frac {9}{2}}}{8} - \frac {74235 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {12005 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {184877 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} \]
-81*(1 - 2*x)**(15/2)/64 + 81*(1 - 2*x)**(13/2)/4 - 97335*(1 - 2*x)**(11/2 )/704 + 4165*(1 - 2*x)**(9/2)/8 - 74235*(1 - 2*x)**(7/2)/64 + 12005*(1 - 2 *x)**(5/2)/8 - 184877*(1 - 2*x)**(3/2)/192
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=-\frac {81}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {81}{4} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {97335}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {4165}{8} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {74235}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12005}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {184877}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
-81/64*(-2*x + 1)^(15/2) + 81/4*(-2*x + 1)^(13/2) - 97335/704*(-2*x + 1)^( 11/2) + 4165/8*(-2*x + 1)^(9/2) - 74235/64*(-2*x + 1)^(7/2) + 12005/8*(-2* x + 1)^(5/2) - 184877/192*(-2*x + 1)^(3/2)
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=\frac {81}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {81}{4} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {97335}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {4165}{8} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {74235}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12005}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {184877}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
81/64*(2*x - 1)^7*sqrt(-2*x + 1) + 81/4*(2*x - 1)^6*sqrt(-2*x + 1) + 97335 /704*(2*x - 1)^5*sqrt(-2*x + 1) + 4165/8*(2*x - 1)^4*sqrt(-2*x + 1) + 7423 5/64*(2*x - 1)^3*sqrt(-2*x + 1) + 12005/8*(2*x - 1)^2*sqrt(-2*x + 1) - 184 877/192*(-2*x + 1)^(3/2)
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx=\frac {12005\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {184877\,{\left (1-2\,x\right )}^{3/2}}{192}-\frac {74235\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {4165\,{\left (1-2\,x\right )}^{9/2}}{8}-\frac {97335\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {81\,{\left (1-2\,x\right )}^{13/2}}{4}-\frac {81\,{\left (1-2\,x\right )}^{15/2}}{64} \]