Integrand size = 22, antiderivative size = 66 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=-\frac {9317}{80} (1-2 x)^{5/2}+\frac {8349}{56} (1-2 x)^{7/2}-\frac {935}{12} (1-2 x)^{9/2}+\frac {1675}{88} (1-2 x)^{11/2}-\frac {375}{208} (1-2 x)^{13/2} \] Output:
-9317/80*(1-2*x)^(5/2)+8349/56*(1-2*x)^(7/2)-935/12*(1-2*x)^(9/2)+1675/88* (1-2*x)^(11/2)-375/208*(1-2*x)^(13/2)
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=-\frac {(1-2 x)^{5/2} \left (421301+1295695 x+1899800 x^2+1420125 x^3+433125 x^4\right )}{15015} \] Input:
Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^3,x]
Output:
-1/15015*((1 - 2*x)^(5/2)*(421301 + 1295695*x + 1899800*x^2 + 1420125*x^3 + 433125*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int (1-2 x)^{3/2} (3 x+2) (5 x+3)^3 \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {375}{16} (1-2 x)^{11/2}-\frac {1675}{8} (1-2 x)^{9/2}+\frac {2805}{4} (1-2 x)^{7/2}-\frac {8349}{8} (1-2 x)^{5/2}+\frac {9317}{16} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {375}{208} (1-2 x)^{13/2}+\frac {1675}{88} (1-2 x)^{11/2}-\frac {935}{12} (1-2 x)^{9/2}+\frac {8349}{56} (1-2 x)^{7/2}-\frac {9317}{80} (1-2 x)^{5/2}\) |
Input:
Int[(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^3,x]
Output:
(-9317*(1 - 2*x)^(5/2))/80 + (8349*(1 - 2*x)^(7/2))/56 - (935*(1 - 2*x)^(9 /2))/12 + (1675*(1 - 2*x)^(11/2))/88 - (375*(1 - 2*x)^(13/2))/208
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 = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (433125 x^{4}+1420125 x^{3}+1899800 x^{2}+1295695 x +421301\right )}{15015}\) | \(30\) |
pseudoelliptic | \(-\frac {1500 \left (x -\frac {1}{2}\right )^{2} \left (x^{4}+\frac {541}{165} x^{3}+\frac {10856}{2475} x^{2}+\frac {259139}{86625} x +\frac {421301}{433125}\right ) \sqrt {1-2 x}}{13}\) | \(33\) |
orering | \(\frac {\left (-1+2 x \right ) \left (433125 x^{4}+1420125 x^{3}+1899800 x^{2}+1295695 x +421301\right ) \left (1-2 x \right )^{\frac {3}{2}}}{15015}\) | \(35\) |
trager | \(\left (-\frac {1500}{13} x^{6}-\frac {37600}{143} x^{5}-\frac {67195}{429} x^{4}+\frac {199259}{3003} x^{3}+\frac {532592}{5005} x^{2}+\frac {389509}{15015} x -\frac {421301}{15015}\right ) \sqrt {1-2 x}\) | \(39\) |
risch | \(\frac {\left (1732500 x^{6}+3948000 x^{5}+2351825 x^{4}-996295 x^{3}-1597776 x^{2}-389509 x +421301\right ) \left (-1+2 x \right )}{15015 \sqrt {1-2 x}}\) | \(45\) |
derivativedivides | \(-\frac {9317 \left (1-2 x \right )^{\frac {5}{2}}}{80}+\frac {8349 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {935 \left (1-2 x \right )^{\frac {9}{2}}}{12}+\frac {1675 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {375 \left (1-2 x \right )^{\frac {13}{2}}}{208}\) | \(47\) |
default | \(-\frac {9317 \left (1-2 x \right )^{\frac {5}{2}}}{80}+\frac {8349 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {935 \left (1-2 x \right )^{\frac {9}{2}}}{12}+\frac {1675 \left (1-2 x \right )^{\frac {11}{2}}}{88}-\frac {375 \left (1-2 x \right )^{\frac {13}{2}}}{208}\) | \(47\) |
meijerg | \(-\frac {81 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {351 \sqrt {\pi }}{35}-\frac {351 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{280}}{\sqrt {\pi }}-\frac {2565 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {370 \sqrt {\pi }}{231}-\frac {185 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{14784}}{\sqrt {\pi }}-\frac {1125 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}\) | \(217\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/15015*(1-2*x)^(5/2)*(433125*x^4+1420125*x^3+1899800*x^2+1295695*x+42130 1)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=-\frac {1}{15015} \, {\left (1732500 \, x^{6} + 3948000 \, x^{5} + 2351825 \, x^{4} - 996295 \, x^{3} - 1597776 \, x^{2} - 389509 \, x + 421301\right )} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="fricas")
Output:
-1/15015*(1732500*x^6 + 3948000*x^5 + 2351825*x^4 - 996295*x^3 - 1597776*x ^2 - 389509*x + 421301)*sqrt(-2*x + 1)
Time = 1.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=- \frac {375 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} + \frac {1675 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {935 \left (1 - 2 x\right )^{\frac {9}{2}}}{12} + \frac {8349 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {9317 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)*(3+5*x)**3,x)
Output:
-375*(1 - 2*x)**(13/2)/208 + 1675*(1 - 2*x)**(11/2)/88 - 935*(1 - 2*x)**(9 /2)/12 + 8349*(1 - 2*x)**(7/2)/56 - 9317*(1 - 2*x)**(5/2)/80
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=-\frac {375}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {1675}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {935}{12} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8349}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {9317}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="maxima")
Output:
-375/208*(-2*x + 1)^(13/2) + 1675/88*(-2*x + 1)^(11/2) - 935/12*(-2*x + 1) ^(9/2) + 8349/56*(-2*x + 1)^(7/2) - 9317/80*(-2*x + 1)^(5/2)
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=-\frac {375}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {1675}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {935}{12} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8349}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {9317}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x, algorithm="giac")
Output:
-375/208*(2*x - 1)^6*sqrt(-2*x + 1) - 1675/88*(2*x - 1)^5*sqrt(-2*x + 1) - 935/12*(2*x - 1)^4*sqrt(-2*x + 1) - 8349/56*(2*x - 1)^3*sqrt(-2*x + 1) - 9317/80*(2*x - 1)^2*sqrt(-2*x + 1)
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=\frac {8349\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {9317\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {935\,{\left (1-2\,x\right )}^{9/2}}{12}+\frac {1675\,{\left (1-2\,x\right )}^{11/2}}{88}-\frac {375\,{\left (1-2\,x\right )}^{13/2}}{208} \] Input:
int((1 - 2*x)^(3/2)*(3*x + 2)*(5*x + 3)^3,x)
Output:
(8349*(1 - 2*x)^(7/2))/56 - (9317*(1 - 2*x)^(5/2))/80 - (935*(1 - 2*x)^(9/ 2))/12 + (1675*(1 - 2*x)^(11/2))/88 - (375*(1 - 2*x)^(13/2))/208
Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^3 \, dx=\frac {\sqrt {-2 x +1}\, \left (-1732500 x^{6}-3948000 x^{5}-2351825 x^{4}+996295 x^{3}+1597776 x^{2}+389509 x -421301\right )}{15015} \] Input:
int((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^3,x)
Output:
(sqrt( - 2*x + 1)*( - 1732500*x**6 - 3948000*x**5 - 2351825*x**4 + 996295* x**3 + 1597776*x**2 + 389509*x - 421301))/15015