Integrand size = 22, antiderivative size = 105 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {1294139}{640} (1-2 x)^{5/2}+\frac {559433}{128} (1-2 x)^{7/2}-\frac {564235}{128} (1-2 x)^{9/2}+\frac {3658095 (1-2 x)^{11/2}}{1408}-\frac {1580985 (1-2 x)^{13/2}}{1664}+\frac {136647}{640} (1-2 x)^{15/2}-\frac {59049 (1-2 x)^{17/2}}{2176}+\frac {3645 (1-2 x)^{19/2}}{2432} \]
-1294139/640*(1-2*x)^(5/2)+559433/128*(1-2*x)^(7/2)-564235/128*(1-2*x)^(9/ 2)+3658095/1408*(1-2*x)^(11/2)-1580985/1664*(1-2*x)^(13/2)+136647/640*(1-2 *x)^(15/2)-59049/2176*(1-2*x)^(17/2)+3645/2432*(1-2*x)^(19/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {(1-2 x)^{5/2} \left (51677856+214047840 x+512679760 x^2+817490880 x^3+876286620 x^4+607227192 x^5+246022920 x^6+44304975 x^7\right )}{230945} \]
-1/230945*((1 - 2*x)^(5/2)*(51677856 + 214047840*x + 512679760*x^2 + 81749 0880*x^3 + 876286620*x^4 + 607227192*x^5 + 246022920*x^6 + 44304975*x^7))
Time = 0.21 (sec) , antiderivative size = 105, 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.
\(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^6 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {3645}{128} (1-2 x)^{17/2}+\frac {59049}{128} (1-2 x)^{15/2}-\frac {409941}{128} (1-2 x)^{13/2}+\frac {1580985}{128} (1-2 x)^{11/2}-\frac {3658095}{128} (1-2 x)^{9/2}+\frac {5078115}{128} (1-2 x)^{7/2}-\frac {3916031}{128} (1-2 x)^{5/2}+\frac {1294139}{128} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3645 (1-2 x)^{19/2}}{2432}-\frac {59049 (1-2 x)^{17/2}}{2176}+\frac {136647}{640} (1-2 x)^{15/2}-\frac {1580985 (1-2 x)^{13/2}}{1664}+\frac {3658095 (1-2 x)^{11/2}}{1408}-\frac {564235}{128} (1-2 x)^{9/2}+\frac {559433}{128} (1-2 x)^{7/2}-\frac {1294139}{640} (1-2 x)^{5/2}\) |
(-1294139*(1 - 2*x)^(5/2))/640 + (559433*(1 - 2*x)^(7/2))/128 - (564235*(1 - 2*x)^(9/2))/128 + (3658095*(1 - 2*x)^(11/2))/1408 - (1580985*(1 - 2*x)^ (13/2))/1664 + (136647*(1 - 2*x)^(15/2))/640 - (59049*(1 - 2*x)^(17/2))/21 76 + (3645*(1 - 2*x)^(19/2))/2432
3.19.57.3.1 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 = 1.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (44304975 x^{7}+246022920 x^{6}+607227192 x^{5}+876286620 x^{4}+817490880 x^{3}+512679760 x^{2}+214047840 x +51677856\right )}{230945}\) | \(45\) |
trager | \(\left (-\frac {14580}{19} x^{9}-\frac {1128492}{323} x^{8}-\frac {10413441}{1615} x^{7}-\frac {120205512}{20995} x^{6}-\frac {372044232}{230945} x^{5}+\frac {68591572}{46189} x^{4}+\frac {75407360}{46189} x^{3}+\frac {136800176}{230945} x^{2}-\frac {7336416}{230945} x -\frac {51677856}{230945}\right ) \sqrt {1-2 x}\) | \(54\) |
pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (177219900 x^{9}+806871780 x^{8}+1489122063 x^{7}+1322260632 x^{6}+372044232 x^{5}-342957860 x^{4}-377036800 x^{3}-136800176 x^{2}+7336416 x +51677856\right )}{230945}\) | \(55\) |
risch | \(\frac {\left (177219900 x^{9}+806871780 x^{8}+1489122063 x^{7}+1322260632 x^{6}+372044232 x^{5}-342957860 x^{4}-377036800 x^{3}-136800176 x^{2}+7336416 x +51677856\right ) \left (-1+2 x \right )}{230945 \sqrt {1-2 x}}\) | \(60\) |
derivativedivides | \(-\frac {1294139 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {559433 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {564235 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {3658095 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {1580985 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {136647 \left (1-2 x \right )^{\frac {15}{2}}}{640}-\frac {59049 \left (1-2 x \right )^{\frac {17}{2}}}{2176}+\frac {3645 \left (1-2 x \right )^{\frac {19}{2}}}{2432}\) | \(74\) |
default | \(-\frac {1294139 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {559433 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {564235 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {3658095 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {1580985 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {136647 \left (1-2 x \right )^{\frac {15}{2}}}{640}-\frac {59049 \left (1-2 x \right )^{\frac {17}{2}}}{2176}+\frac {3645 \left (1-2 x \right )^{\frac {19}{2}}}{2432}\) | \(74\) |
meijerg | \(-\frac {72 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {2048 \sqrt {\pi }}{35}-\frac {256 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{35}}{\sqrt {\pi }}-\frac {1755 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{2 \sqrt {\pi }}+\frac {\frac {288 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{28}}{\sqrt {\pi }}-\frac {27135 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{32 \sqrt {\pi }}+\frac {\frac {29376 \sqrt {\pi }}{5005}-\frac {459 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{80080}}{\sqrt {\pi }}-\frac {50301 \left (-\frac {8192 \sqrt {\pi }}{765765}+\frac {4 \sqrt {\pi }\, \left (7687680 x^{8}-4612608 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{765765}\right )}{512 \sqrt {\pi }}+\frac {\frac {3888 \sqrt {\pi }}{46189}-\frac {243 \sqrt {\pi }\, \left (298721280 x^{9}-175718400 x^{8}+878592 x^{7}+473088 x^{6}+258048 x^{5}+143360 x^{4}+81920 x^{3}+49152 x^{2}+32768 x +32768\right ) \sqrt {1-2 x}}{94595072}}{\sqrt {\pi }}\) | \(406\) |
-1/230945*(1-2*x)^(5/2)*(44304975*x^7+246022920*x^6+607227192*x^5+87628662 0*x^4+817490880*x^3+512679760*x^2+214047840*x+51677856)
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {1}{230945} \, {\left (177219900 \, x^{9} + 806871780 \, x^{8} + 1489122063 \, x^{7} + 1322260632 \, x^{6} + 372044232 \, x^{5} - 342957860 \, x^{4} - 377036800 \, x^{3} - 136800176 \, x^{2} + 7336416 \, x + 51677856\right )} \sqrt {-2 \, x + 1} \]
-1/230945*(177219900*x^9 + 806871780*x^8 + 1489122063*x^7 + 1322260632*x^6 + 372044232*x^5 - 342957860*x^4 - 377036800*x^3 - 136800176*x^2 + 7336416 *x + 51677856)*sqrt(-2*x + 1)
Time = 0.89 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=\frac {3645 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} - \frac {59049 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} + \frac {136647 \left (1 - 2 x\right )^{\frac {15}{2}}}{640} - \frac {1580985 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} + \frac {3658095 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {564235 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {559433 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} - \frac {1294139 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} \]
3645*(1 - 2*x)**(19/2)/2432 - 59049*(1 - 2*x)**(17/2)/2176 + 136647*(1 - 2 *x)**(15/2)/640 - 1580985*(1 - 2*x)**(13/2)/1664 + 3658095*(1 - 2*x)**(11/ 2)/1408 - 564235*(1 - 2*x)**(9/2)/128 + 559433*(1 - 2*x)**(7/2)/128 - 1294 139*(1 - 2*x)**(5/2)/640
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=\frac {3645}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} - \frac {59049}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {136647}{640} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {1580985}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {3658095}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {564235}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {559433}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1294139}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]
3645/2432*(-2*x + 1)^(19/2) - 59049/2176*(-2*x + 1)^(17/2) + 136647/640*(- 2*x + 1)^(15/2) - 1580985/1664*(-2*x + 1)^(13/2) + 3658095/1408*(-2*x + 1) ^(11/2) - 564235/128*(-2*x + 1)^(9/2) + 559433/128*(-2*x + 1)^(7/2) - 1294 139/640*(-2*x + 1)^(5/2)
Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=-\frac {3645}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} - \frac {59049}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {136647}{640} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {1580985}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {3658095}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {564235}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {559433}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1294139}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]
-3645/2432*(2*x - 1)^9*sqrt(-2*x + 1) - 59049/2176*(2*x - 1)^8*sqrt(-2*x + 1) - 136647/640*(2*x - 1)^7*sqrt(-2*x + 1) - 1580985/1664*(2*x - 1)^6*sqr t(-2*x + 1) - 3658095/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 564235/128*(2*x - 1)^4*sqrt(-2*x + 1) - 559433/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1294139/640* (2*x - 1)^2*sqrt(-2*x + 1)
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^6 (3+5 x) \, dx=\frac {559433\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {1294139\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {564235\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {3658095\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {1580985\,{\left (1-2\,x\right )}^{13/2}}{1664}+\frac {136647\,{\left (1-2\,x\right )}^{15/2}}{640}-\frac {59049\,{\left (1-2\,x\right )}^{17/2}}{2176}+\frac {3645\,{\left (1-2\,x\right )}^{19/2}}{2432} \]