Integrand size = 24, antiderivative size = 105 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {2033647}{384 (1-2 x)^{3/2}}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095}{128} \sqrt {1-2 x}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {6075 (1-2 x)^{11/2}}{1408} \]
2033647/384/(1-2*x)^(3/2)+1965635/128*(1-2*x)^(3/2)-514017/128*(1-2*x)^(5/ 2)+672003/896*(1-2*x)^(7/2)-10845/128*(1-2*x)^(9/2)+6075/1408*(1-2*x)^(11/ 2)-6206585/128/(1-2*x)^(1/2)-8117095/128*(1-2*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {21852008-65622552 x+32450916 x^2+9702012 x^3+5121279 x^4+2456001 x^5+806085 x^6+127575 x^7}{231 (1-2 x)^{3/2}} \]
-1/231*(21852008 - 65622552*x + 32450916*x^2 + 9702012*x^3 + 5121279*x^4 + 2456001*x^5 + 806085*x^6 + 127575*x^7)/(1 - 2*x)^(3/2)
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {(3 x+2)^5 (5 x+3)^2}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6075}{128} (1-2 x)^{9/2}+\frac {97605}{128} (1-2 x)^{7/2}-\frac {672003}{128} (1-2 x)^{5/2}+\frac {2570085}{128} (1-2 x)^{3/2}-\frac {5896905}{128} \sqrt {1-2 x}+\frac {8117095}{128 \sqrt {1-2 x}}-\frac {6206585}{128 (1-2 x)^{3/2}}+\frac {2033647}{128 (1-2 x)^{5/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6075 (1-2 x)^{11/2}}{1408}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {8117095}{128} \sqrt {1-2 x}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {2033647}{384 (1-2 x)^{3/2}}\) |
2033647/(384*(1 - 2*x)^(3/2)) - 6206585/(128*Sqrt[1 - 2*x]) - (8117095*Sqr t[1 - 2*x])/128 + (1965635*(1 - 2*x)^(3/2))/128 - (514017*(1 - 2*x)^(5/2)) /128 + (672003*(1 - 2*x)^(7/2))/896 - (10845*(1 - 2*x)^(9/2))/128 + (6075* (1 - 2*x)^(11/2))/1408
3.22.46.3.1 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 = 3.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008}{231 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
pseudoelliptic | \(\frac {-127575 x^{7}-806085 x^{6}-2456001 x^{5}-5121279 x^{4}-9702012 x^{3}-32450916 x^{2}+65622552 x -21852008}{231 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(-\frac {\left (127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008\right ) \sqrt {1-2 x}}{231 \left (-1+2 x \right )^{2}}\) | \(52\) |
risch | \(\frac {127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}+32450916 x^{2}-65622552 x +21852008}{231 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(52\) |
derivativedivides | \(\frac {2033647}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1965635 \left (1-2 x \right )^{\frac {3}{2}}}{128}-\frac {514017 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {10845 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {6075 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095 \sqrt {1-2 x}}{128}\) | \(74\) |
default | \(\frac {2033647}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1965635 \left (1-2 x \right )^{\frac {3}{2}}}{128}-\frac {514017 \left (1-2 x \right )^{\frac {5}{2}}}{128}+\frac {672003 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {10845 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {6075 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095 \sqrt {1-2 x}}{128}\) | \(74\) |
meijerg | \(-\frac {192 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {1040 \sqrt {\pi }-\frac {130 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {7240 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {24880 \sqrt {\pi }-\frac {1555 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {9615 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }}+\frac {20376 \sqrt {\pi }-\frac {2547 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{128 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {2295 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {10800 \sqrt {\pi }}{11}-\frac {675 \sqrt {\pi }\, \left (18432 x^{7}+14336 x^{6}+12288 x^{5}+12288 x^{4}+16384 x^{3}+49152 x^{2}-98304 x +32768\right )}{22528 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\) | \(324\) |
-1/231/(1-2*x)^(3/2)*(127575*x^7+806085*x^6+2456001*x^5+5121279*x^4+970201 2*x^3+32450916*x^2-65622552*x+21852008)
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {{\left (127575 \, x^{7} + 806085 \, x^{6} + 2456001 \, x^{5} + 5121279 \, x^{4} + 9702012 \, x^{3} + 32450916 \, x^{2} - 65622552 \, x + 21852008\right )} \sqrt {-2 \, x + 1}}{231 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/231*(127575*x^7 + 806085*x^6 + 2456001*x^5 + 5121279*x^4 + 9702012*x^3 + 32450916*x^2 - 65622552*x + 21852008)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)
Time = 1.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {6075 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {10845 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {672003 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {514017 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} + \frac {1965635 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} - \frac {8117095 \sqrt {1 - 2 x}}{128} - \frac {6206585}{128 \sqrt {1 - 2 x}} + \frac {2033647}{384 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
6075*(1 - 2*x)**(11/2)/1408 - 10845*(1 - 2*x)**(9/2)/128 + 672003*(1 - 2*x )**(7/2)/896 - 514017*(1 - 2*x)**(5/2)/128 + 1965635*(1 - 2*x)**(3/2)/128 - 8117095*sqrt(1 - 2*x)/128 - 6206585/(128*sqrt(1 - 2*x)) + 2033647/(384*( 1 - 2*x)**(3/2))
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {6075}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {10845}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {672003}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {514017}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} + \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
6075/1408*(-2*x + 1)^(11/2) - 10845/128*(-2*x + 1)^(9/2) + 672003/896*(-2* x + 1)^(7/2) - 514017/128*(-2*x + 1)^(5/2) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) + 26411/192*(705*x - 314)/(-2*x + 1)^(3/2)
Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {6075}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {10845}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {672003}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {514017}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} - \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
-6075/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 10845/128*(2*x - 1)^4*sqrt(-2*x + 1) - 672003/896*(2*x - 1)^3*sqrt(-2*x + 1) - 514017/128*(2*x - 1)^2*sqrt(- 2*x + 1) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) - 264 11/192*(705*x - 314)/((2*x - 1)*sqrt(-2*x + 1))
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {\frac {6206585\,x}{64}-\frac {4146527}{96}}{{\left (1-2\,x\right )}^{3/2}}-\frac {8117095\,\sqrt {1-2\,x}}{128}+\frac {1965635\,{\left (1-2\,x\right )}^{3/2}}{128}-\frac {514017\,{\left (1-2\,x\right )}^{5/2}}{128}+\frac {672003\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {10845\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {6075\,{\left (1-2\,x\right )}^{11/2}}{1408} \]