Integrand size = 22, antiderivative size = 79 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=-\frac {26411}{96} (1-2 x)^{3/2}+\frac {57281}{160} (1-2 x)^{5/2}-\frac {3549}{16} (1-2 x)^{7/2}+\frac {1197}{16} (1-2 x)^{9/2}-\frac {4671}{352} (1-2 x)^{11/2}+\frac {405}{416} (1-2 x)^{13/2} \]
-26411/96*(1-2*x)^(3/2)+57281/160*(1-2*x)^(5/2)-3549/16*(1-2*x)^(7/2)+1197 /16*(1-2*x)^(9/2)-4671/352*(1-2*x)^(11/2)+405/416*(1-2*x)^(13/2)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=-\frac {(1-2 x)^{3/2} \left (163888+388704 x+577080 x^2+540000 x^3+288360 x^4+66825 x^5\right )}{2145} \]
-1/2145*((1 - 2*x)^(3/2)*(163888 + 388704*x + 577080*x^2 + 540000*x^3 + 28 8360*x^4 + 66825*x^5))
Time = 0.18 (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.
\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^4 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (-\frac {405}{32} (1-2 x)^{11/2}+\frac {4671}{32} (1-2 x)^{9/2}-\frac {10773}{16} (1-2 x)^{7/2}+\frac {24843}{16} (1-2 x)^{5/2}-\frac {57281}{32} (1-2 x)^{3/2}+\frac {26411}{32} \sqrt {1-2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {405}{416} (1-2 x)^{13/2}-\frac {4671}{352} (1-2 x)^{11/2}+\frac {1197}{16} (1-2 x)^{9/2}-\frac {3549}{16} (1-2 x)^{7/2}+\frac {57281}{160} (1-2 x)^{5/2}-\frac {26411}{96} (1-2 x)^{3/2}\) |
(-26411*(1 - 2*x)^(3/2))/96 + (57281*(1 - 2*x)^(5/2))/160 - (3549*(1 - 2*x )^(7/2))/16 + (1197*(1 - 2*x)^(9/2))/16 - (4671*(1 - 2*x)^(11/2))/352 + (4 05*(1 - 2*x)^(13/2))/416
3.18.92.3.1 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 = 35, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (66825 x^{5}+288360 x^{4}+540000 x^{3}+577080 x^{2}+388704 x +163888\right )}{2145}\) | \(35\) |
trager | \(\left (\frac {810}{13} x^{6}+\frac {33993}{143} x^{5}+\frac {52776}{143} x^{4}+\frac {40944}{143} x^{3}+\frac {66776}{715} x^{2}-\frac {60928}{2145} x -\frac {163888}{2145}\right ) \sqrt {1-2 x}\) | \(39\) |
pseudoelliptic | \(\frac {\sqrt {1-2 x}\, \left (133650 x^{6}+509895 x^{5}+791640 x^{4}+614160 x^{3}+200328 x^{2}-60928 x -163888\right )}{2145}\) | \(40\) |
risch | \(-\frac {\left (133650 x^{6}+509895 x^{5}+791640 x^{4}+614160 x^{3}+200328 x^{2}-60928 x -163888\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\) | \(45\) |
derivativedivides | \(-\frac {26411 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {57281 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {3549 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1197 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {4671 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {405 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
default | \(-\frac {26411 \left (1-2 x \right )^{\frac {3}{2}}}{96}+\frac {57281 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {3549 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1197 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {4671 \left (1-2 x \right )^{\frac {11}{2}}}{352}+\frac {405 \left (1-2 x \right )^{\frac {13}{2}}}{416}\) | \(56\) |
meijerg | \(\frac {16 \sqrt {\pi }-8 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {46 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {752 \sqrt {\pi }}{35}-\frac {94 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{35}}{\sqrt {\pi }}-\frac {54 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{\sqrt {\pi }}+\frac {\frac {168 \sqrt {\pi }}{55}-\frac {21 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{880}}{\sqrt {\pi }}-\frac {405 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{128 \sqrt {\pi }}\) | \(220\) |
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=\frac {1}{2145} \, {\left (133650 \, x^{6} + 509895 \, x^{5} + 791640 \, x^{4} + 614160 \, x^{3} + 200328 \, x^{2} - 60928 \, x - 163888\right )} \sqrt {-2 \, x + 1} \]
1/2145*(133650*x^6 + 509895*x^5 + 791640*x^4 + 614160*x^3 + 200328*x^2 - 6 0928*x - 163888)*sqrt(-2*x + 1)
Time = 0.68 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=\frac {405 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {4671 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {1197 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {3549 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} + \frac {57281 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {26411 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \]
405*(1 - 2*x)**(13/2)/416 - 4671*(1 - 2*x)**(11/2)/352 + 1197*(1 - 2*x)**( 9/2)/16 - 3549*(1 - 2*x)**(7/2)/16 + 57281*(1 - 2*x)**(5/2)/160 - 26411*(1 - 2*x)**(3/2)/96
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=\frac {405}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {4671}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1197}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3549}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {57281}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {26411}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
405/416*(-2*x + 1)^(13/2) - 4671/352*(-2*x + 1)^(11/2) + 1197/16*(-2*x + 1 )^(9/2) - 3549/16*(-2*x + 1)^(7/2) + 57281/160*(-2*x + 1)^(5/2) - 26411/96 *(-2*x + 1)^(3/2)
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=\frac {405}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {4671}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1197}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3549}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {57281}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {26411}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
405/416*(2*x - 1)^6*sqrt(-2*x + 1) + 4671/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1197/16*(2*x - 1)^4*sqrt(-2*x + 1) + 3549/16*(2*x - 1)^3*sqrt(-2*x + 1) + 57281/160*(2*x - 1)^2*sqrt(-2*x + 1) - 26411/96*(-2*x + 1)^(3/2)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x) \, dx=\frac {57281\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {26411\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {3549\,{\left (1-2\,x\right )}^{7/2}}{16}+\frac {1197\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {4671\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {405\,{\left (1-2\,x\right )}^{13/2}}{416} \]