Integrand size = 22, antiderivative size = 66 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=-\frac {539}{16} (1-2 x)^{7/2}+\frac {3283}{72} (1-2 x)^{9/2}-\frac {1071}{44} (1-2 x)^{11/2}+\frac {621}{104} (1-2 x)^{13/2}-\frac {9}{16} (1-2 x)^{15/2} \] Output:
-539/16*(1-2*x)^(7/2)+3283/72*(1-2*x)^(9/2)-1071/44*(1-2*x)^(11/2)+621/104 *(1-2*x)^(13/2)-9/16*(1-2*x)^(15/2)
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=-\frac {(1-2 x)^{7/2} \left (9038+32378 x+50463 x^2+38313 x^3+11583 x^4\right )}{1287} \] Input:
Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x),x]
Output:
-1/1287*((1 - 2*x)^(7/2)*(9038 + 32378*x + 50463*x^2 + 38313*x^3 + 11583*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.
\(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^3 (5 x+3) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {135}{16} (1-2 x)^{13/2}-\frac {621}{8} (1-2 x)^{11/2}+\frac {1071}{4} (1-2 x)^{9/2}-\frac {3283}{8} (1-2 x)^{7/2}+\frac {3773}{16} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9}{16} (1-2 x)^{15/2}+\frac {621}{104} (1-2 x)^{13/2}-\frac {1071}{44} (1-2 x)^{11/2}+\frac {3283}{72} (1-2 x)^{9/2}-\frac {539}{16} (1-2 x)^{7/2}\) |
Input:
Int[(1 - 2*x)^(5/2)*(2 + 3*x)^3*(3 + 5*x),x]
Output:
(-539*(1 - 2*x)^(7/2))/16 + (3283*(1 - 2*x)^(9/2))/72 - (1071*(1 - 2*x)^(1 1/2))/44 + (621*(1 - 2*x)^(13/2))/104 - (9*(1 - 2*x)^(15/2))/16
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 = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45
method | result | size |
gosper | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (11583 x^{4}+38313 x^{3}+50463 x^{2}+32378 x +9038\right )}{1287}\) | \(30\) |
pseudoelliptic | \(-\frac {\left (1-2 x \right )^{\frac {7}{2}} \left (11583 x^{4}+38313 x^{3}+50463 x^{2}+32378 x +9038\right )}{1287}\) | \(30\) |
orering | \(\frac {\left (-1+2 x \right ) \left (11583 x^{4}+38313 x^{3}+50463 x^{2}+32378 x +9038\right ) \left (1-2 x \right )^{\frac {5}{2}}}{1287}\) | \(35\) |
trager | \(\left (72 x^{7}+\frac {1692}{13} x^{6}+\frac {1494}{143} x^{5}-\frac {128237}{1287} x^{4}-\frac {51767}{1287} x^{3}+\frac {11783}{429} x^{2}+\frac {21850}{1287} x -\frac {9038}{1287}\right ) \sqrt {1-2 x}\) | \(44\) |
derivativedivides | \(-\frac {539 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {3283 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {1071 \left (1-2 x \right )^{\frac {11}{2}}}{44}+\frac {621 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {9 \left (1-2 x \right )^{\frac {15}{2}}}{16}\) | \(47\) |
default | \(-\frac {539 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {3283 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {1071 \left (1-2 x \right )^{\frac {11}{2}}}{44}+\frac {621 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {9 \left (1-2 x \right )^{\frac {15}{2}}}{16}\) | \(47\) |
risch | \(-\frac {\left (92664 x^{7}+167508 x^{6}+13446 x^{5}-128237 x^{4}-51767 x^{3}+35349 x^{2}+21850 x -9038\right ) \left (-1+2 x \right )}{1287 \sqrt {1-2 x}}\) | \(50\) |
meijerg | \(\frac {\frac {24 \sqrt {\pi }}{7}-\frac {12 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {555 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{8 \sqrt {\pi }}+\frac {\frac {76 \sqrt {\pi }}{77}-\frac {19 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{308}}{\sqrt {\pi }}-\frac {5265 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {24 \sqrt {\pi }}{1001}-\frac {3 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{32032}}{\sqrt {\pi }}\) | \(242\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x),x,method=_RETURNVERBOSE)
Output:
-1/1287*(1-2*x)^(7/2)*(11583*x^4+38313*x^3+50463*x^2+32378*x+9038)
Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=\frac {1}{1287} \, {\left (92664 \, x^{7} + 167508 \, x^{6} + 13446 \, x^{5} - 128237 \, x^{4} - 51767 \, x^{3} + 35349 \, x^{2} + 21850 \, x - 9038\right )} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x),x, algorithm="fricas")
Output:
1/1287*(92664*x^7 + 167508*x^6 + 13446*x^5 - 128237*x^4 - 51767*x^3 + 3534 9*x^2 + 21850*x - 9038)*sqrt(-2*x + 1)
Time = 0.80 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=- \frac {9 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} + \frac {621 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {1071 \left (1 - 2 x\right )^{\frac {11}{2}}}{44} + \frac {3283 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} - \frac {539 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**3*(3+5*x),x)
Output:
-9*(1 - 2*x)**(15/2)/16 + 621*(1 - 2*x)**(13/2)/104 - 1071*(1 - 2*x)**(11/ 2)/44 + 3283*(1 - 2*x)**(9/2)/72 - 539*(1 - 2*x)**(7/2)/16
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=-\frac {9}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {621}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1071}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3283}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {539}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x),x, algorithm="maxima")
Output:
-9/16*(-2*x + 1)^(15/2) + 621/104*(-2*x + 1)^(13/2) - 1071/44*(-2*x + 1)^( 11/2) + 3283/72*(-2*x + 1)^(9/2) - 539/16*(-2*x + 1)^(7/2)
Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=\frac {9}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {621}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1071}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3283}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {539}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x),x, algorithm="giac")
Output:
9/16*(2*x - 1)^7*sqrt(-2*x + 1) + 621/104*(2*x - 1)^6*sqrt(-2*x + 1) + 107 1/44*(2*x - 1)^5*sqrt(-2*x + 1) + 3283/72*(2*x - 1)^4*sqrt(-2*x + 1) + 539 /16*(2*x - 1)^3*sqrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=\frac {3283\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {539\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {1071\,{\left (1-2\,x\right )}^{11/2}}{44}+\frac {621\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {9\,{\left (1-2\,x\right )}^{15/2}}{16} \] Input:
int((1 - 2*x)^(5/2)*(3*x + 2)^3*(5*x + 3),x)
Output:
(3283*(1 - 2*x)^(9/2))/72 - (539*(1 - 2*x)^(7/2))/16 - (1071*(1 - 2*x)^(11 /2))/44 + (621*(1 - 2*x)^(13/2))/104 - (9*(1 - 2*x)^(15/2))/16
Time = 0.15 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x) \, dx=\frac {\sqrt {-2 x +1}\, \left (92664 x^{7}+167508 x^{6}+13446 x^{5}-128237 x^{4}-51767 x^{3}+35349 x^{2}+21850 x -9038\right )}{1287} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^3*(3+5*x),x)
Output:
(sqrt( - 2*x + 1)*(92664*x**7 + 167508*x**6 + 13446*x**5 - 128237*x**4 - 5 1767*x**3 + 35349*x**2 + 21850*x - 9038))/1287