Integrand size = 24, antiderivative size = 105 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585}{128} \sqrt {1-2 x}-\frac {8117095}{384} (1-2 x)^{3/2}+\frac {1179381}{128} (1-2 x)^{5/2}-\frac {367155}{128} (1-2 x)^{7/2}+\frac {74667}{128} (1-2 x)^{9/2}-\frac {97605 (1-2 x)^{11/2}}{1408}+\frac {6075 (1-2 x)^{13/2}}{1664} \] Output:
2033647/128/(1-2*x)^(1/2)+6206585/128*(1-2*x)^(1/2)-8117095/384*(1-2*x)^(3 /2)+1179381/128*(1-2*x)^(5/2)-367155/128*(1-2*x)^(7/2)+74667/128*(1-2*x)^( 9/2)-97605/1408*(1-2*x)^(11/2)+6075/1664*(1-2*x)^(13/2)
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {21493640-21370088 x-10015804 x^2-7945164 x^3-5928885 x^4-3350637 x^5-1201635 x^6-200475 x^7}{429 \sqrt {1-2 x}} \] Input:
Integrate[((2 + 3*x)^5*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]
Output:
(21493640 - 21370088*x - 10015804*x^2 - 7945164*x^3 - 5928885*x^4 - 335063 7*x^5 - 1201635*x^6 - 200475*x^7)/(429*Sqrt[1 - 2*x])
Time = 0.23 (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.
\(\displaystyle \int \frac {(3 x+2)^5 (5 x+3)^2}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6075}{128} (1-2 x)^{11/2}+\frac {97605}{128} (1-2 x)^{9/2}-\frac {672003}{128} (1-2 x)^{7/2}+\frac {2570085}{128} (1-2 x)^{5/2}-\frac {5896905}{128} (1-2 x)^{3/2}+\frac {8117095}{128} \sqrt {1-2 x}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {2033647}{128 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6075 (1-2 x)^{13/2}}{1664}-\frac {97605 (1-2 x)^{11/2}}{1408}+\frac {74667}{128} (1-2 x)^{9/2}-\frac {367155}{128} (1-2 x)^{7/2}+\frac {1179381}{128} (1-2 x)^{5/2}-\frac {8117095}{384} (1-2 x)^{3/2}+\frac {6206585}{128} \sqrt {1-2 x}+\frac {2033647}{128 \sqrt {1-2 x}}\) |
Input:
Int[((2 + 3*x)^5*(3 + 5*x)^2)/(1 - 2*x)^(3/2),x]
Output:
2033647/(128*Sqrt[1 - 2*x]) + (6206585*Sqrt[1 - 2*x])/128 - (8117095*(1 - 2*x)^(3/2))/384 + (1179381*(1 - 2*x)^(5/2))/128 - (367155*(1 - 2*x)^(7/2)) /128 + (74667*(1 - 2*x)^(9/2))/128 - (97605*(1 - 2*x)^(11/2))/1408 + (6075 *(1 - 2*x)^(13/2))/1664
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 = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
risch | \(-\frac {200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(\frac {-200475 x^{7}-1201635 x^{6}-3350637 x^{5}-5928885 x^{4}-7945164 x^{3}-10015804 x^{2}-21370088 x +21493640}{429 \sqrt {1-2 x}}\) | \(45\) |
orering | \(\frac {\left (-1+2 x \right ) \left (200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640\right )}{429 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(50\) |
trager | \(\frac {\left (200475 x^{7}+1201635 x^{6}+3350637 x^{5}+5928885 x^{4}+7945164 x^{3}+10015804 x^{2}+21370088 x -21493640\right ) \sqrt {1-2 x}}{-429+858 x}\) | \(52\) |
derivativedivides | \(\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585 \sqrt {1-2 x}}{128}-\frac {8117095 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {1179381 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {367155 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {74667 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {97605 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {6075 \left (1-2 x \right )^{\frac {13}{2}}}{1664}\) | \(74\) |
default | \(\frac {2033647}{128 \sqrt {1-2 x}}+\frac {6206585 \sqrt {1-2 x}}{128}-\frac {8117095 \left (1-2 x \right )^{\frac {3}{2}}}{384}+\frac {1179381 \left (1-2 x \right )^{\frac {5}{2}}}{128}-\frac {367155 \left (1-2 x \right )^{\frac {7}{2}}}{128}+\frac {74667 \left (1-2 x \right )^{\frac {9}{2}}}{128}-\frac {97605 \left (1-2 x \right )^{\frac {11}{2}}}{1408}+\frac {6075 \left (1-2 x \right )^{\frac {13}{2}}}{1664}\) | \(74\) |
meijerg | \(-\frac {288 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-3120 \sqrt {\pi }+\frac {390 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3620 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-14928 \sqrt {\pi }+\frac {933 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {28845 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{8 \sqrt {\pi }}+\frac {-6792 \sqrt {\pi }+\frac {849 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{128 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {6885 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {32400 \sqrt {\pi }}{143}+\frac {2025 \sqrt {\pi }\, \left (-67584 x^{7}-43008 x^{6}-28672 x^{5}-20480 x^{4}-16384 x^{3}-16384 x^{2}-32768 x +32768\right )}{292864 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(324\) |
Input:
int((2+3*x)^5*(3+5*x)^2/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/429/(1-2*x)^(1/2)*(200475*x^7+1201635*x^6+3350637*x^5+5928885*x^4+79451 64*x^3+10015804*x^2+21370088*x-21493640)
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {{\left (200475 \, x^{7} + 1201635 \, x^{6} + 3350637 \, x^{5} + 5928885 \, x^{4} + 7945164 \, x^{3} + 10015804 \, x^{2} + 21370088 \, x - 21493640\right )} \sqrt {-2 \, x + 1}}{429 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^5*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
1/429*(200475*x^7 + 1201635*x^6 + 3350637*x^5 + 5928885*x^4 + 7945164*x^3 + 10015804*x^2 + 21370088*x - 21493640)*sqrt(-2*x + 1)/(2*x - 1)
Time = 1.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {97605 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {74667 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {367155 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {1179381 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} - \frac {8117095 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} + \frac {6206585 \sqrt {1 - 2 x}}{128} + \frac {2033647}{128 \sqrt {1 - 2 x}} \] Input:
integrate((2+3*x)**5*(3+5*x)**2/(1-2*x)**(3/2),x)
Output:
6075*(1 - 2*x)**(13/2)/1664 - 97605*(1 - 2*x)**(11/2)/1408 + 74667*(1 - 2* x)**(9/2)/128 - 367155*(1 - 2*x)**(7/2)/128 + 1179381*(1 - 2*x)**(5/2)/128 - 8117095*(1 - 2*x)**(3/2)/384 + 6206585*sqrt(1 - 2*x)/128 + 2033647/(128 *sqrt(1 - 2*x))
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {97605}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {74667}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {367155}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {1179381}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8117095}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6206585}{128} \, \sqrt {-2 \, x + 1} + \frac {2033647}{128 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^5*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
6075/1664*(-2*x + 1)^(13/2) - 97605/1408*(-2*x + 1)^(11/2) + 74667/128*(-2 *x + 1)^(9/2) - 367155/128*(-2*x + 1)^(7/2) + 1179381/128*(-2*x + 1)^(5/2) - 8117095/384*(-2*x + 1)^(3/2) + 6206585/128*sqrt(-2*x + 1) + 2033647/128 /sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {6075}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {97605}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {74667}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {367155}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {1179381}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8117095}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6206585}{128} \, \sqrt {-2 \, x + 1} + \frac {2033647}{128 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^5*(3+5*x)^2/(1-2*x)^(3/2),x, algorithm="giac")
Output:
6075/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 97605/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 74667/128*(2*x - 1)^4*sqrt(-2*x + 1) + 367155/128*(2*x - 1)^3*sqrt(-2 *x + 1) + 1179381/128*(2*x - 1)^2*sqrt(-2*x + 1) - 8117095/384*(-2*x + 1)^ (3/2) + 6206585/128*sqrt(-2*x + 1) + 2033647/128/sqrt(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {2033647}{128\,\sqrt {1-2\,x}}+\frac {6206585\,\sqrt {1-2\,x}}{128}-\frac {8117095\,{\left (1-2\,x\right )}^{3/2}}{384}+\frac {1179381\,{\left (1-2\,x\right )}^{5/2}}{128}-\frac {367155\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {74667\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {97605\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {6075\,{\left (1-2\,x\right )}^{13/2}}{1664} \] Input:
int(((3*x + 2)^5*(5*x + 3)^2)/(1 - 2*x)^(3/2),x)
Output:
2033647/(128*(1 - 2*x)^(1/2)) + (6206585*(1 - 2*x)^(1/2))/128 - (8117095*( 1 - 2*x)^(3/2))/384 + (1179381*(1 - 2*x)^(5/2))/128 - (367155*(1 - 2*x)^(7 /2))/128 + (74667*(1 - 2*x)^(9/2))/128 - (97605*(1 - 2*x)^(11/2))/1408 + ( 6075*(1 - 2*x)^(13/2))/1664
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx=\frac {-200475 x^{7}-1201635 x^{6}-3350637 x^{5}-5928885 x^{4}-7945164 x^{3}-10015804 x^{2}-21370088 x +21493640}{429 \sqrt {-2 x +1}} \] Input:
int((2+3*x)^5*(3+5*x)^2/(1-2*x)^(3/2),x)
Output:
( - 200475*x**7 - 1201635*x**6 - 3350637*x**5 - 5928885*x**4 - 7945164*x** 3 - 10015804*x**2 - 21370088*x + 21493640)/(429*sqrt( - 2*x + 1))