Integrand size = 24, antiderivative size = 66 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {847}{16} (1-2 x)^{7/2}+\frac {1309}{18} (1-2 x)^{9/2}-\frac {3467}{88} (1-2 x)^{11/2}+\frac {255}{26} (1-2 x)^{13/2}-\frac {15}{16} (1-2 x)^{15/2} \] Output:
-847/16*(1-2*x)^(7/2)+1309/18*(1-2*x)^(9/2)-3467/88*(1-2*x)^(11/2)+255/26* (1-2*x)^(13/2)-15/16*(1-2*x)^(15/2)
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {(1-2 x)^{7/2} \left (13826+50450 x+80307 x^2+62370 x^3+19305 x^4\right )}{1287} \] Input:
Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^2,x]
Output:
-1/1287*((1 - 2*x)^(7/2)*(13826 + 50450*x + 80307*x^2 + 62370*x^3 + 19305* x^4))
Time = 0.18 (sec) , antiderivative size = 66, 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.
\(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^2 \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {225}{16} (1-2 x)^{13/2}-\frac {255}{2} (1-2 x)^{11/2}+\frac {3467}{8} (1-2 x)^{9/2}-\frac {1309}{2} (1-2 x)^{7/2}+\frac {5929}{16} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {15}{16} (1-2 x)^{15/2}+\frac {255}{26} (1-2 x)^{13/2}-\frac {3467}{88} (1-2 x)^{11/2}+\frac {1309}{18} (1-2 x)^{9/2}-\frac {847}{16} (1-2 x)^{7/2}\) |
Input:
Int[(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^2,x]
Output:
(-847*(1 - 2*x)^(7/2))/16 + (1309*(1 - 2*x)^(9/2))/18 - (3467*(1 - 2*x)^(1 1/2))/88 + (255*(1 - 2*x)^(13/2))/26 - (15*(1 - 2*x)^(15/2))/16
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 = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (19305 x^{4}+62370 x^{3}+80307 x^{2}+50450 x +13826\right )}{1287}\) | \(30\) |
pseudoelliptic | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (19305 x^{4}+62370 x^{3}+80307 x^{2}+50450 x +13826\right )}{1287}\) | \(30\) |
orering | \(\frac {\left (-1+2 x \right ) \left (19305 x^{4}+62370 x^{3}+80307 x^{2}+50450 x +13826\right ) \left (1-2 x \right )^{\frac {5}{2}}}{1287}\) | \(35\) |
trager | \(\left (120 x^{7}+\frac {2700}{13} x^{6}+\frac {1094}{143} x^{5}-\frac {205169}{1287} x^{4}-\frac {75320}{1287} x^{3}+\frac {18827}{429} x^{2}+\frac {32506}{1287} x -\frac {13826}{1287}\right ) \sqrt {1-2 x}\) | \(44\) |
derivativedivides | \(-\frac {847 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1309 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {3467 \left (1-2 x \right )^{\frac {11}{2}}}{88}+\frac {255 \left (1-2 x \right )^{\frac {13}{2}}}{26}-\frac {15 \left (1-2 x \right )^{\frac {15}{2}}}{16}\) | \(47\) |
default | \(-\frac {847 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {1309 \left (1-2 x \right )^{\frac {9}{2}}}{18}-\frac {3467 \left (1-2 x \right )^{\frac {11}{2}}}{88}+\frac {255 \left (1-2 x \right )^{\frac {13}{2}}}{26}-\frac {15 \left (1-2 x \right )^{\frac {15}{2}}}{16}\) | \(47\) |
risch | \(-\frac {\left (154440 x^{7}+267300 x^{6}+9846 x^{5}-205169 x^{4}-75320 x^{3}+56481 x^{2}+32506 x -13826\right ) \left (-1+2 x \right )}{1287 \sqrt {1-2 x}}\) | \(50\) |
meijerg | \(\frac {\frac {36 \sqrt {\pi }}{7}-\frac {18 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {855 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{8 \sqrt {\pi }}+\frac {\frac {1082 \sqrt {\pi }}{693}-\frac {541 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{5544}}{\sqrt {\pi }}-\frac {4275 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{64 \sqrt {\pi }}+\frac {\frac {40 \sqrt {\pi }}{1001}-\frac {5 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{32032}}{\sqrt {\pi }}\) | \(242\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/1287*(1-2*x)^(7/2)*(19305*x^4+62370*x^3+80307*x^2+50450*x+13826)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {1}{1287} \, {\left (154440 \, x^{7} + 267300 \, x^{6} + 9846 \, x^{5} - 205169 \, x^{4} - 75320 \, x^{3} + 56481 \, x^{2} + 32506 \, x - 13826\right )} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^2,x, algorithm="fricas")
Output:
1/1287*(154440*x^7 + 267300*x^6 + 9846*x^5 - 205169*x^4 - 75320*x^3 + 5648 1*x^2 + 32506*x - 13826)*sqrt(-2*x + 1)
Time = 0.80 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=- \frac {15 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} + \frac {255 \left (1 - 2 x\right )^{\frac {13}{2}}}{26} - \frac {3467 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} + \frac {1309 \left (1 - 2 x\right )^{\frac {9}{2}}}{18} - \frac {847 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**2*(3+5*x)**2,x)
Output:
-15*(1 - 2*x)**(15/2)/16 + 255*(1 - 2*x)**(13/2)/26 - 3467*(1 - 2*x)**(11/ 2)/88 + 1309*(1 - 2*x)**(9/2)/18 - 847*(1 - 2*x)**(7/2)/16
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=-\frac {15}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {255}{26} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {3467}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1309}{18} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {847}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^2,x, algorithm="maxima")
Output:
-15/16*(-2*x + 1)^(15/2) + 255/26*(-2*x + 1)^(13/2) - 3467/88*(-2*x + 1)^( 11/2) + 1309/18*(-2*x + 1)^(9/2) - 847/16*(-2*x + 1)^(7/2)
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {15}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {255}{26} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {3467}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1309}{18} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {847}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^2,x, algorithm="giac")
Output:
15/16*(2*x - 1)^7*sqrt(-2*x + 1) + 255/26*(2*x - 1)^6*sqrt(-2*x + 1) + 346 7/88*(2*x - 1)^5*sqrt(-2*x + 1) + 1309/18*(2*x - 1)^4*sqrt(-2*x + 1) + 847 /16*(2*x - 1)^3*sqrt(-2*x + 1)
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {1309\,{\left (1-2\,x\right )}^{9/2}}{18}-\frac {847\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {3467\,{\left (1-2\,x\right )}^{11/2}}{88}+\frac {255\,{\left (1-2\,x\right )}^{13/2}}{26}-\frac {15\,{\left (1-2\,x\right )}^{15/2}}{16} \] Input:
int((1 - 2*x)^(5/2)*(3*x + 2)^2*(5*x + 3)^2,x)
Output:
(1309*(1 - 2*x)^(9/2))/18 - (847*(1 - 2*x)^(7/2))/16 - (3467*(1 - 2*x)^(11 /2))/88 + (255*(1 - 2*x)^(13/2))/26 - (15*(1 - 2*x)^(15/2))/16
Time = 0.15 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx=\frac {\sqrt {-2 x +1}\, \left (154440 x^{7}+267300 x^{6}+9846 x^{5}-205169 x^{4}-75320 x^{3}+56481 x^{2}+32506 x -13826\right )}{1287} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^2,x)
Output:
(sqrt( - 2*x + 1)*(154440*x**7 + 267300*x**6 + 9846*x**5 - 205169*x**4 - 7 5320*x**3 + 56481*x**2 + 32506*x - 13826))/1287