Integrand size = 22, antiderivative size = 92 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=-\frac {184877}{320} (1-2 x)^{5/2}+\frac {8575}{8} (1-2 x)^{7/2}-\frac {173215}{192} (1-2 x)^{9/2}+\frac {37485}{88} (1-2 x)^{11/2}-\frac {97335}{832} (1-2 x)^{13/2}+\frac {351}{20} (1-2 x)^{15/2}-\frac {1215 (1-2 x)^{17/2}}{1088} \]
-184877/320*(1-2*x)^(5/2)+8575/8*(1-2*x)^(7/2)-173215/192*(1-2*x)^(9/2)+37 485/88*(1-2*x)^(11/2)-97335/832*(1-2*x)^(13/2)+351/20*(1-2*x)^(15/2)-1215/ 1088*(1-2*x)^(17/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=-\frac {(1-2 x)^{5/2} \left (3012632+11562520 x+24424220 x^2+32431860 x^3+26832465 x^4+12660219 x^5+2606175 x^6\right )}{36465} \]
-1/36465*((1 - 2*x)^(5/2)*(3012632 + 11562520*x + 24424220*x^2 + 32431860* x^3 + 26832465*x^4 + 12660219*x^5 + 2606175*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.
\(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^5 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {1215}{64} (1-2 x)^{15/2}-\frac {1053}{4} (1-2 x)^{13/2}+\frac {97335}{64} (1-2 x)^{11/2}-\frac {37485}{8} (1-2 x)^{9/2}+\frac {519645}{64} (1-2 x)^{7/2}-\frac {60025}{8} (1-2 x)^{5/2}+\frac {184877}{64} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1215 (1-2 x)^{17/2}}{1088}+\frac {351}{20} (1-2 x)^{15/2}-\frac {97335}{832} (1-2 x)^{13/2}+\frac {37485}{88} (1-2 x)^{11/2}-\frac {173215}{192} (1-2 x)^{9/2}+\frac {8575}{8} (1-2 x)^{7/2}-\frac {184877}{320} (1-2 x)^{5/2}\) |
(-184877*(1 - 2*x)^(5/2))/320 + (8575*(1 - 2*x)^(7/2))/8 - (173215*(1 - 2* x)^(9/2))/192 + (37485*(1 - 2*x)^(11/2))/88 - (97335*(1 - 2*x)^(13/2))/832 + (351*(1 - 2*x)^(15/2))/20 - (1215*(1 - 2*x)^(17/2))/1088
3.19.58.3.1 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 = 0.97 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (2606175 x^{6}+12660219 x^{5}+26832465 x^{4}+32431860 x^{3}+24424220 x^{2}+11562520 x +3012632\right )}{36465}\) | \(40\) |
trager | \(\left (-\frac {4860}{17} x^{8}-\frac {93744}{85} x^{7}-\frac {1796823}{1105} x^{6}-\frac {11685933}{12155} x^{5}+\frac {1039619}{7293} x^{4}+\frac {3802988}{7293} x^{3}+\frac {3258444}{12155} x^{2}+\frac {488008}{36465} x -\frac {3012632}{36465}\right ) \sqrt {1-2 x}\) | \(49\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (10424700 x^{8}+40216176 x^{7}+59295159 x^{6}+35057799 x^{5}-5198095 x^{4}-19014940 x^{3}-9775332 x^{2}-488008 x +3012632\right )}{36465}\) | \(50\) |
risch | \(\frac {\left (10424700 x^{8}+40216176 x^{7}+59295159 x^{6}+35057799 x^{5}-5198095 x^{4}-19014940 x^{3}-9775332 x^{2}-488008 x +3012632\right ) \left (-1+2 x \right )}{36465 \sqrt {1-2 x}}\) | \(55\) |
derivativedivides | \(-\frac {184877 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {8575 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {173215 \left (1-2 x \right )^{\frac {9}{2}}}{192}+\frac {37485 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {97335 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {351 \left (1-2 x \right )^{\frac {15}{2}}}{20}-\frac {1215 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\) | \(65\) |
default | \(-\frac {184877 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {8575 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {173215 \left (1-2 x \right )^{\frac {9}{2}}}{192}+\frac {37485 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {97335 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {351 \left (1-2 x \right )^{\frac {15}{2}}}{20}-\frac {1215 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\) | \(65\) |
meijerg | \(-\frac {36 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {176 \sqrt {\pi }}{7}-\frac {22 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {315 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{\sqrt {\pi }}+\frac {\frac {912 \sqrt {\pi }}{77}-\frac {57 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{616}}{\sqrt {\pi }}-\frac {11745 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{64 \sqrt {\pi }}+\frac {\frac {4248 \sqrt {\pi }}{5005}-\frac {531 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{640640}}{\sqrt {\pi }}-\frac {3645 \left (-\frac {8192 \sqrt {\pi }}{765765}+\frac {4 \sqrt {\pi }\, \left (7687680 x^{8}-4612608 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{765765}\right )}{512 \sqrt {\pi }}\) | \(338\) |
-1/36465*(1-2*x)^(5/2)*(2606175*x^6+12660219*x^5+26832465*x^4+32431860*x^3 +24424220*x^2+11562520*x+3012632)
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=-\frac {1}{36465} \, {\left (10424700 \, x^{8} + 40216176 \, x^{7} + 59295159 \, x^{6} + 35057799 \, x^{5} - 5198095 \, x^{4} - 19014940 \, x^{3} - 9775332 \, x^{2} - 488008 \, x + 3012632\right )} \sqrt {-2 \, x + 1} \]
-1/36465*(10424700*x^8 + 40216176*x^7 + 59295159*x^6 + 35057799*x^5 - 5198 095*x^4 - 19014940*x^3 - 9775332*x^2 - 488008*x + 3012632)*sqrt(-2*x + 1)
Time = 0.84 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=- \frac {1215 \left (1 - 2 x\right )^{\frac {17}{2}}}{1088} + \frac {351 \left (1 - 2 x\right )^{\frac {15}{2}}}{20} - \frac {97335 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {37485 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {173215 \left (1 - 2 x\right )^{\frac {9}{2}}}{192} + \frac {8575 \left (1 - 2 x\right )^{\frac {7}{2}}}{8} - \frac {184877 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} \]
-1215*(1 - 2*x)**(17/2)/1088 + 351*(1 - 2*x)**(15/2)/20 - 97335*(1 - 2*x)* *(13/2)/832 + 37485*(1 - 2*x)**(11/2)/88 - 173215*(1 - 2*x)**(9/2)/192 + 8 575*(1 - 2*x)**(7/2)/8 - 184877*(1 - 2*x)**(5/2)/320
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=-\frac {1215}{1088} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {351}{20} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {97335}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {37485}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {173215}{192} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8575}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {184877}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]
-1215/1088*(-2*x + 1)^(17/2) + 351/20*(-2*x + 1)^(15/2) - 97335/832*(-2*x + 1)^(13/2) + 37485/88*(-2*x + 1)^(11/2) - 173215/192*(-2*x + 1)^(9/2) + 8 575/8*(-2*x + 1)^(7/2) - 184877/320*(-2*x + 1)^(5/2)
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=-\frac {1215}{1088} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {351}{20} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {97335}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {37485}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {173215}{192} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8575}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {184877}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]
-1215/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 351/20*(2*x - 1)^7*sqrt(-2*x + 1) - 97335/832*(2*x - 1)^6*sqrt(-2*x + 1) - 37485/88*(2*x - 1)^5*sqrt(-2*x + 1) - 173215/192*(2*x - 1)^4*sqrt(-2*x + 1) - 8575/8*(2*x - 1)^3*sqrt(-2*x + 1) - 184877/320*(2*x - 1)^2*sqrt(-2*x + 1)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^5 (3+5 x) \, dx=\frac {8575\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {184877\,{\left (1-2\,x\right )}^{5/2}}{320}-\frac {173215\,{\left (1-2\,x\right )}^{9/2}}{192}+\frac {37485\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {97335\,{\left (1-2\,x\right )}^{13/2}}{832}+\frac {351\,{\left (1-2\,x\right )}^{15/2}}{20}-\frac {1215\,{\left (1-2\,x\right )}^{17/2}}{1088} \]