Integrand size = 22, antiderivative size = 79 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=-\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2} \]
-26411/160*(1-2*x)^(5/2)+8183/32*(1-2*x)^(7/2)-8281/48*(1-2*x)^(9/2)+10773 /176*(1-2*x)^(11/2)-4671/416*(1-2*x)^(13/2)+27/32*(1-2*x)^(15/2)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=-\frac {(1-2 x)^{5/2} \left (66592+230000 x+410320 x^2+424440 x^3+240570 x^4+57915 x^5\right )}{2145} \]
-1/2145*((1 - 2*x)^(5/2)*(66592 + 230000*x + 410320*x^2 + 424440*x^3 + 240 570*x^4 + 57915*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^4 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {405}{32} (1-2 x)^{13/2}+\frac {4671}{32} (1-2 x)^{11/2}-\frac {10773}{16} (1-2 x)^{9/2}+\frac {24843}{16} (1-2 x)^{7/2}-\frac {57281}{32} (1-2 x)^{5/2}+\frac {26411}{32} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {27}{32} (1-2 x)^{15/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {26411}{160} (1-2 x)^{5/2}\) |
(-26411*(1 - 2*x)^(5/2))/160 + (8183*(1 - 2*x)^(7/2))/32 - (8281*(1 - 2*x) ^(9/2))/48 + (10773*(1 - 2*x)^(11/2))/176 - (4671*(1 - 2*x)^(13/2))/416 + (27*(1 - 2*x)^(15/2))/32
3.19.59.3.1 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 = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (57915 x^{5}+240570 x^{4}+424440 x^{3}+410320 x^{2}+230000 x +66592\right )}{2145}\) | \(35\) |
trager | \(\left (-108 x^{7}-\frac {4428}{13} x^{6}-\frac {52893}{143} x^{5}-\frac {36818}{429} x^{4}+\frac {59368}{429} x^{3}+\frac {81104}{715} x^{2}+\frac {36368}{2145} x -\frac {66592}{2145}\right ) \sqrt {1-2 x}\) | \(44\) |
pseudoelliptic | \(-\frac {\left (231660 x^{7}+730620 x^{6}+793395 x^{5}+184090 x^{4}-296840 x^{3}-243312 x^{2}-36368 x +66592\right ) \sqrt {1-2 x}}{2145}\) | \(45\) |
risch | \(\frac {\left (231660 x^{7}+730620 x^{6}+793395 x^{5}+184090 x^{4}-296840 x^{3}-243312 x^{2}-36368 x +66592\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\) | \(50\) |
derivativedivides | \(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\) | \(56\) |
default | \(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\) | \(56\) |
meijerg | \(-\frac {18 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {368 \sqrt {\pi }}{35}-\frac {46 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{35}}{\sqrt {\pi }}-\frac {423 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{4 \sqrt {\pi }}+\frac {\frac {1152 \sqrt {\pi }}{385}-\frac {9 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{385}}{\sqrt {\pi }}-\frac {3969 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {72 \sqrt {\pi }}{1001}-\frac {9 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{128128}}{\sqrt {\pi }}\) | \(275\) |
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=-\frac {1}{2145} \, {\left (231660 \, x^{7} + 730620 \, x^{6} + 793395 \, x^{5} + 184090 \, x^{4} - 296840 \, x^{3} - 243312 \, x^{2} - 36368 \, x + 66592\right )} \sqrt {-2 \, x + 1} \]
-1/2145*(231660*x^7 + 730620*x^6 + 793395*x^5 + 184090*x^4 - 296840*x^3 - 243312*x^2 - 36368*x + 66592)*sqrt(-2*x + 1)
Time = 0.76 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=\frac {27 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} + \frac {10773 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {8281 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {8183 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {26411 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} \]
27*(1 - 2*x)**(15/2)/32 - 4671*(1 - 2*x)**(13/2)/416 + 10773*(1 - 2*x)**(1 1/2)/176 - 8281*(1 - 2*x)**(9/2)/48 + 8183*(1 - 2*x)**(7/2)/32 - 26411*(1 - 2*x)**(5/2)/160
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=\frac {27}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4671}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {10773}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {8281}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8183}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26411}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]
27/32*(-2*x + 1)^(15/2) - 4671/416*(-2*x + 1)^(13/2) + 10773/176*(-2*x + 1 )^(11/2) - 8281/48*(-2*x + 1)^(9/2) + 8183/32*(-2*x + 1)^(7/2) - 26411/160 *(-2*x + 1)^(5/2)
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=-\frac {27}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4671}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {10773}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {8281}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8183}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26411}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]
-27/32*(2*x - 1)^7*sqrt(-2*x + 1) - 4671/416*(2*x - 1)^6*sqrt(-2*x + 1) - 10773/176*(2*x - 1)^5*sqrt(-2*x + 1) - 8281/48*(2*x - 1)^4*sqrt(-2*x + 1) - 8183/32*(2*x - 1)^3*sqrt(-2*x + 1) - 26411/160*(2*x - 1)^2*sqrt(-2*x + 1 )
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx=\frac {8183\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {26411\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {8281\,{\left (1-2\,x\right )}^{9/2}}{48}+\frac {10773\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {4671\,{\left (1-2\,x\right )}^{13/2}}{416}+\frac {27\,{\left (1-2\,x\right )}^{15/2}}{32} \]