Integrand size = 24, antiderivative size = 105 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {3195731}{640} (1-2 x)^{5/2}+\frac {1405173}{128} (1-2 x)^{7/2}-\frac {4324397}{384} (1-2 x)^{9/2}+\frac {9504551 (1-2 x)^{11/2}}{1408}-\frac {4177401 (1-2 x)^{13/2}}{1664}+\frac {73431}{128} (1-2 x)^{15/2}-\frac {161325 (1-2 x)^{17/2}}{2176}+\frac {10125 (1-2 x)^{19/2}}{2432} \] Output:
-3195731/640*(1-2*x)^(5/2)+1405173/128*(1-2*x)^(7/2)-4324397/384*(1-2*x)^( 9/2)+9504551/1408*(1-2*x)^(11/2)-4177401/1664*(1-2*x)^(13/2)+73431/128*(1- 2*x)^(15/2)-161325/2176*(1-2*x)^(17/2)+10125/2432*(1-2*x)^(19/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {(1-2 x)^{5/2} \left (369438704+1547888800 x+3771434840 x^2+6142984080 x^3+6744559140 x^4+4795033815 x^5+1995171750 x^6+369208125 x^7\right )}{692835} \] Input:
Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^3,x]
Output:
-1/692835*((1 - 2*x)^(5/2)*(369438704 + 1547888800*x + 3771434840*x^2 + 61 42984080*x^3 + 6744559140*x^4 + 4795033815*x^5 + 1995171750*x^6 + 36920812 5*x^7))
Time = 0.21 (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.
| \(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^4 (5 x+3)^3 \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {10125}{128} (1-2 x)^{17/2}+\frac {161325}{128} (1-2 x)^{15/2}-\frac {1101465}{128} (1-2 x)^{13/2}+\frac {4177401}{128} (1-2 x)^{11/2}-\frac {9504551}{128} (1-2 x)^{9/2}+\frac {12973191}{128} (1-2 x)^{7/2}-\frac {9836211}{128} (1-2 x)^{5/2}+\frac {3195731}{128} (1-2 x)^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {10125 (1-2 x)^{19/2}}{2432}-\frac {161325 (1-2 x)^{17/2}}{2176}+\frac {73431}{128} (1-2 x)^{15/2}-\frac {4177401 (1-2 x)^{13/2}}{1664}+\frac {9504551 (1-2 x)^{11/2}}{1408}-\frac {4324397}{384} (1-2 x)^{9/2}+\frac {1405173}{128} (1-2 x)^{7/2}-\frac {3195731}{640} (1-2 x)^{5/2}\) |
Input:
Int[(1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^3,x]
Output:
(-3195731*(1 - 2*x)^(5/2))/640 + (1405173*(1 - 2*x)^(7/2))/128 - (4324397* (1 - 2*x)^(9/2))/384 + (9504551*(1 - 2*x)^(11/2))/1408 - (4177401*(1 - 2*x )^(13/2))/1664 + (73431*(1 - 2*x)^(15/2))/128 - (161325*(1 - 2*x)^(17/2))/ 2176 + (10125*(1 - 2*x)^(19/2))/2432
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 = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
| method | result | size |
| gosper | \(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (369208125 x^{7}+1995171750 x^{6}+4795033815 x^{5}+6744559140 x^{4}+6142984080 x^{3}+3771434840 x^{2}+1547888800 x +369438704\right )}{692835}\) | \(45\) |
| orering | \(\frac {\left (-1+2 x \right ) \left (369208125 x^{7}+1995171750 x^{6}+4795033815 x^{5}+6744559140 x^{4}+6142984080 x^{3}+3771434840 x^{2}+1547888800 x +369438704\right ) \left (1-2 x \right )^{\frac {3}{2}}}{692835}\) | \(50\) |
| trager | \(\left (-\frac {40500}{19} x^{9}-\frac {3032100}{323} x^{8}-\frac {5393313}{323} x^{7}-\frac {59353170}{4199} x^{6}-\frac {159248905}{46189} x^{5}+\frac {548327564}{138567} x^{4}+\frac {550240016}{138567} x^{3}+\frac {314121848}{230945} x^{2}-\frac {70133984}{692835} x -\frac {369438704}{692835}\right ) \sqrt {1-2 x}\) | \(54\) |
| pseudoelliptic | \(-\frac {\sqrt {1-2 x}\, \left (1476832500 x^{9}+6503854500 x^{8}+11568656385 x^{7}+9793273050 x^{6}+2388733575 x^{5}-2741637820 x^{4}-2751200080 x^{3}-942365544 x^{2}+70133984 x +369438704\right )}{692835}\) | \(55\) |
| risch | \(\frac {\left (1476832500 x^{9}+6503854500 x^{8}+11568656385 x^{7}+9793273050 x^{6}+2388733575 x^{5}-2741637820 x^{4}-2751200080 x^{3}-942365544 x^{2}+70133984 x +369438704\right ) \left (-1+2 x \right )}{692835 \sqrt {1-2 x}}\) | \(60\) |
| derivativedivides | \(-\frac {3195731 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {1405173 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {4324397 \left (1-2 x \right )^{\frac {9}{2}}}{384}+\frac {9504551 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {4177401 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {73431 \left (1-2 x \right )^{\frac {15}{2}}}{128}-\frac {161325 \left (1-2 x \right )^{\frac {17}{2}}}{2176}+\frac {10125 \left (1-2 x \right )^{\frac {19}{2}}}{2432}\) | \(74\) |
| default | \(-\frac {3195731 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {1405173 \left (1-2 x \right )^{\frac {7}{2}}}{128}-\frac {4324397 \left (1-2 x \right )^{\frac {9}{2}}}{384}+\frac {9504551 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {4177401 \left (1-2 x \right )^{\frac {13}{2}}}{1664}+\frac {73431 \left (1-2 x \right )^{\frac {15}{2}}}{128}-\frac {161325 \left (1-2 x \right )^{\frac {17}{2}}}{2176}+\frac {10125 \left (1-2 x \right )^{\frac {19}{2}}}{2432}\) | \(74\) |
| meijerg | \(-\frac {162 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {4752 \sqrt {\pi }}{35}-\frac {594 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{35}}{\sqrt {\pi }}-\frac {8397 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{4 \sqrt {\pi }}+\frac {\frac {117184 \sqrt {\pi }}{1155}-\frac {1831 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{2310}}{\sqrt {\pi }}-\frac {275841 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {15384 \sqrt {\pi }}{1001}-\frac {1923 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{128128}}{\sqrt {\pi }}-\frac {135675 \left (-\frac {8192 \sqrt {\pi }}{765765}+\frac {4 \sqrt {\pi }\, \left (7687680 x^{8}-4612608 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{765765}\right )}{512 \sqrt {\pi }}+\frac {\frac {10800 \sqrt {\pi }}{46189}-\frac {675 \sqrt {\pi }\, \left (298721280 x^{9}-175718400 x^{8}+878592 x^{7}+473088 x^{6}+258048 x^{5}+143360 x^{4}+81920 x^{3}+49152 x^{2}+32768 x +32768\right ) \sqrt {1-2 x}}{94595072}}{\sqrt {\pi }}\) | \(406\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)^4*(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/692835*(1-2*x)^(5/2)*(369208125*x^7+1995171750*x^6+4795033815*x^5+67445 59140*x^4+6142984080*x^3+3771434840*x^2+1547888800*x+369438704)
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {1}{692835} \, {\left (1476832500 \, x^{9} + 6503854500 \, x^{8} + 11568656385 \, x^{7} + 9793273050 \, x^{6} + 2388733575 \, x^{5} - 2741637820 \, x^{4} - 2751200080 \, x^{3} - 942365544 \, x^{2} + 70133984 \, x + 369438704\right )} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4*(3+5*x)^3,x, algorithm="fricas")
Output:
-1/692835*(1476832500*x^9 + 6503854500*x^8 + 11568656385*x^7 + 9793273050* x^6 + 2388733575*x^5 - 2741637820*x^4 - 2751200080*x^3 - 942365544*x^2 + 7 0133984*x + 369438704)*sqrt(-2*x + 1)
Time = 0.91 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125 \left (1 - 2 x\right )^{\frac {19}{2}}}{2432} - \frac {161325 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} + \frac {73431 \left (1 - 2 x\right )^{\frac {15}{2}}}{128} - \frac {4177401 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} + \frac {9504551 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {4324397 \left (1 - 2 x\right )^{\frac {9}{2}}}{384} + \frac {1405173 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} - \frac {3195731 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)**4*(3+5*x)**3,x)
Output:
10125*(1 - 2*x)**(19/2)/2432 - 161325*(1 - 2*x)**(17/2)/2176 + 73431*(1 - 2*x)**(15/2)/128 - 4177401*(1 - 2*x)**(13/2)/1664 + 9504551*(1 - 2*x)**(11 /2)/1408 - 4324397*(1 - 2*x)**(9/2)/384 + 1405173*(1 - 2*x)**(7/2)/128 - 3 195731*(1 - 2*x)**(5/2)/640
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {10125}{2432} \, {\left (-2 \, x + 1\right )}^{\frac {19}{2}} - \frac {161325}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {73431}{128} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4177401}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {9504551}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {4324397}{384} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1405173}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3195731}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4*(3+5*x)^3,x, algorithm="maxima")
Output:
10125/2432*(-2*x + 1)^(19/2) - 161325/2176*(-2*x + 1)^(17/2) + 73431/128*( -2*x + 1)^(15/2) - 4177401/1664*(-2*x + 1)^(13/2) + 9504551/1408*(-2*x + 1 )^(11/2) - 4324397/384*(-2*x + 1)^(9/2) + 1405173/128*(-2*x + 1)^(7/2) - 3 195731/640*(-2*x + 1)^(5/2)
Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {10125}{2432} \, {\left (2 \, x - 1\right )}^{9} \sqrt {-2 \, x + 1} - \frac {161325}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {73431}{128} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4177401}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {9504551}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {4324397}{384} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1405173}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3195731}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4*(3+5*x)^3,x, algorithm="giac")
Output:
-10125/2432*(2*x - 1)^9*sqrt(-2*x + 1) - 161325/2176*(2*x - 1)^8*sqrt(-2*x + 1) - 73431/128*(2*x - 1)^7*sqrt(-2*x + 1) - 4177401/1664*(2*x - 1)^6*sq rt(-2*x + 1) - 9504551/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 4324397/384*(2*x - 1)^4*sqrt(-2*x + 1) - 1405173/128*(2*x - 1)^3*sqrt(-2*x + 1) - 3195731/6 40*(2*x - 1)^2*sqrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {1405173\,{\left (1-2\,x\right )}^{7/2}}{128}-\frac {3195731\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {4324397\,{\left (1-2\,x\right )}^{9/2}}{384}+\frac {9504551\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {4177401\,{\left (1-2\,x\right )}^{13/2}}{1664}+\frac {73431\,{\left (1-2\,x\right )}^{15/2}}{128}-\frac {161325\,{\left (1-2\,x\right )}^{17/2}}{2176}+\frac {10125\,{\left (1-2\,x\right )}^{19/2}}{2432} \] Input:
int((1 - 2*x)^(3/2)*(3*x + 2)^4*(5*x + 3)^3,x)
Output:
(1405173*(1 - 2*x)^(7/2))/128 - (3195731*(1 - 2*x)^(5/2))/640 - (4324397*( 1 - 2*x)^(9/2))/384 + (9504551*(1 - 2*x)^(11/2))/1408 - (4177401*(1 - 2*x) ^(13/2))/1664 + (73431*(1 - 2*x)^(15/2))/128 - (161325*(1 - 2*x)^(17/2))/2 176 + (10125*(1 - 2*x)^(19/2))/2432
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^3 \, dx=\frac {\sqrt {-2 x +1}\, \left (-1476832500 x^{9}-6503854500 x^{8}-11568656385 x^{7}-9793273050 x^{6}-2388733575 x^{5}+2741637820 x^{4}+2751200080 x^{3}+942365544 x^{2}-70133984 x -369438704\right )}{692835} \] Input:
int((1-2*x)^(3/2)*(2+3*x)^4*(3+5*x)^3,x)
Output:
(sqrt( - 2*x + 1)*( - 1476832500*x**9 - 6503854500*x**8 - 11568656385*x**7 - 9793273050*x**6 - 2388733575*x**5 + 2741637820*x**4 + 2751200080*x**3 + 942365544*x**2 - 70133984*x - 369438704))/692835