Integrand size = 24, antiderivative size = 92 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {456533}{64 \sqrt {1-2 x}}+\frac {302379}{16} \sqrt {1-2 x}-\frac {444983}{64} (1-2 x)^{3/2}+\frac {98209}{40} (1-2 x)^{5/2}-\frac {260055}{448} (1-2 x)^{7/2}+\frac {1275}{16} (1-2 x)^{9/2}-\frac {3375}{704} (1-2 x)^{11/2} \] Output:
456533/64/(1-2*x)^(1/2)+302379/16*(1-2*x)^(1/2)-444983/64*(1-2*x)^(3/2)+98 209/40*(1-2*x)^(5/2)-260055/448*(1-2*x)^(7/2)+1275/16*(1-2*x)^(9/2)-3375/7 04*(1-2*x)^(11/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {8096086-8012926 x-3611453 x^2-2569643 x^3-1564350 x^4-627375 x^5-118125 x^6}{385 \sqrt {1-2 x}} \] Input:
Integrate[((2 + 3*x)^3*(3 + 5*x)^3)/(1 - 2*x)^(3/2),x]
Output:
(8096086 - 8012926*x - 3611453*x^2 - 2569643*x^3 - 1564350*x^4 - 627375*x^ 5 - 118125*x^6)/(385*Sqrt[1 - 2*x])
Time = 0.23 (sec) , antiderivative size = 92, 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.
\(\displaystyle \int \frac {(3 x+2)^3 (5 x+3)^3}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {3375}{64} (1-2 x)^{9/2}-\frac {11475}{16} (1-2 x)^{7/2}+\frac {260055}{64} (1-2 x)^{5/2}-\frac {98209}{8} (1-2 x)^{3/2}+\frac {1334949}{64} \sqrt {1-2 x}-\frac {302379}{16 \sqrt {1-2 x}}+\frac {456533}{64 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3375}{704} (1-2 x)^{11/2}+\frac {1275}{16} (1-2 x)^{9/2}-\frac {260055}{448} (1-2 x)^{7/2}+\frac {98209}{40} (1-2 x)^{5/2}-\frac {444983}{64} (1-2 x)^{3/2}+\frac {302379}{16} \sqrt {1-2 x}+\frac {456533}{64 \sqrt {1-2 x}}\) |
Input:
Int[((2 + 3*x)^3*(3 + 5*x)^3)/(1 - 2*x)^(3/2),x]
Output:
456533/(64*Sqrt[1 - 2*x]) + (302379*Sqrt[1 - 2*x])/16 - (444983*(1 - 2*x)^ (3/2))/64 + (98209*(1 - 2*x)^(5/2))/40 - (260055*(1 - 2*x)^(7/2))/448 + (1 275*(1 - 2*x)^(9/2))/16 - (3375*(1 - 2*x)^(11/2))/704
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 = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {118125 x^{6}+627375 x^{5}+1564350 x^{4}+2569643 x^{3}+3611453 x^{2}+8012926 x -8096086}{385 \sqrt {1-2 x}}\) | \(40\) |
risch | \(-\frac {118125 x^{6}+627375 x^{5}+1564350 x^{4}+2569643 x^{3}+3611453 x^{2}+8012926 x -8096086}{385 \sqrt {1-2 x}}\) | \(40\) |
pseudoelliptic | \(\frac {-118125 x^{6}-627375 x^{5}-1564350 x^{4}-2569643 x^{3}-3611453 x^{2}-8012926 x +8096086}{385 \sqrt {1-2 x}}\) | \(40\) |
orering | \(\frac {\left (-1+2 x \right ) \left (118125 x^{6}+627375 x^{5}+1564350 x^{4}+2569643 x^{3}+3611453 x^{2}+8012926 x -8096086\right )}{385 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(\frac {\left (118125 x^{6}+627375 x^{5}+1564350 x^{4}+2569643 x^{3}+3611453 x^{2}+8012926 x -8096086\right ) \sqrt {1-2 x}}{-385+770 x}\) | \(47\) |
derivativedivides | \(\frac {456533}{64 \sqrt {1-2 x}}+\frac {302379 \sqrt {1-2 x}}{16}-\frac {444983 \left (1-2 x \right )^{\frac {3}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {260055 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1275 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {11}{2}}}{704}\) | \(65\) |
default | \(\frac {456533}{64 \sqrt {1-2 x}}+\frac {302379 \sqrt {1-2 x}}{16}-\frac {444983 \left (1-2 x \right )^{\frac {3}{2}}}{64}+\frac {98209 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {260055 \left (1-2 x \right )^{\frac {7}{2}}}{448}+\frac {1275 \left (1-2 x \right )^{\frac {9}{2}}}{16}-\frac {3375 \left (1-2 x \right )^{\frac {11}{2}}}{704}\) | \(65\) |
meijerg | \(-\frac {216 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-2052 \sqrt {\pi }+\frac {513 \sqrt {\pi }\, \left (-8 x +8\right )}{2 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {4059 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{2 \sqrt {\pi }}+\frac {-\frac {34238 \sqrt {\pi }}{5}+\frac {17119 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{320 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {20295 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {11400 \sqrt {\pi }}{7}+\frac {1425 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3375 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}\) | \(266\) |
Input:
int((2+3*x)^3*(3+5*x)^3/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/385/(1-2*x)^(1/2)*(118125*x^6+627375*x^5+1564350*x^4+2569643*x^3+361145 3*x^2+8012926*x-8096086)
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {{\left (118125 \, x^{6} + 627375 \, x^{5} + 1564350 \, x^{4} + 2569643 \, x^{3} + 3611453 \, x^{2} + 8012926 \, x - 8096086\right )} \sqrt {-2 \, x + 1}}{385 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^3*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
1/385*(118125*x^6 + 627375*x^5 + 1564350*x^4 + 2569643*x^3 + 3611453*x^2 + 8012926*x - 8096086)*sqrt(-2*x + 1)/(2*x - 1)
Time = 0.92 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=- \frac {3375 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {1275 \left (1 - 2 x\right )^{\frac {9}{2}}}{16} - \frac {260055 \left (1 - 2 x\right )^{\frac {7}{2}}}{448} + \frac {98209 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {444983 \left (1 - 2 x\right )^{\frac {3}{2}}}{64} + \frac {302379 \sqrt {1 - 2 x}}{16} + \frac {456533}{64 \sqrt {1 - 2 x}} \] Input:
integrate((2+3*x)**3*(3+5*x)**3/(1-2*x)**(3/2),x)
Output:
-3375*(1 - 2*x)**(11/2)/704 + 1275*(1 - 2*x)**(9/2)/16 - 260055*(1 - 2*x)* *(7/2)/448 + 98209*(1 - 2*x)**(5/2)/40 - 444983*(1 - 2*x)**(3/2)/64 + 3023 79*sqrt(1 - 2*x)/16 + 456533/(64*sqrt(1 - 2*x))
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=-\frac {3375}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1275}{16} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {260055}{448} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {98209}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {444983}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {302379}{16} \, \sqrt {-2 \, x + 1} + \frac {456533}{64 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^3*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
-3375/704*(-2*x + 1)^(11/2) + 1275/16*(-2*x + 1)^(9/2) - 260055/448*(-2*x + 1)^(7/2) + 98209/40*(-2*x + 1)^(5/2) - 444983/64*(-2*x + 1)^(3/2) + 3023 79/16*sqrt(-2*x + 1) + 456533/64/sqrt(-2*x + 1)
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {3375}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1275}{16} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {260055}{448} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {98209}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {444983}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {302379}{16} \, \sqrt {-2 \, x + 1} + \frac {456533}{64 \, \sqrt {-2 \, x + 1}} \] Input:
integrate((2+3*x)^3*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="giac")
Output:
3375/704*(2*x - 1)^5*sqrt(-2*x + 1) + 1275/16*(2*x - 1)^4*sqrt(-2*x + 1) + 260055/448*(2*x - 1)^3*sqrt(-2*x + 1) + 98209/40*(2*x - 1)^2*sqrt(-2*x + 1) - 444983/64*(-2*x + 1)^(3/2) + 302379/16*sqrt(-2*x + 1) + 456533/64/sqr t(-2*x + 1)
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {456533}{64\,\sqrt {1-2\,x}}+\frac {302379\,\sqrt {1-2\,x}}{16}-\frac {444983\,{\left (1-2\,x\right )}^{3/2}}{64}+\frac {98209\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {260055\,{\left (1-2\,x\right )}^{7/2}}{448}+\frac {1275\,{\left (1-2\,x\right )}^{9/2}}{16}-\frac {3375\,{\left (1-2\,x\right )}^{11/2}}{704} \] Input:
int(((3*x + 2)^3*(5*x + 3)^3)/(1 - 2*x)^(3/2),x)
Output:
456533/(64*(1 - 2*x)^(1/2)) + (302379*(1 - 2*x)^(1/2))/16 - (444983*(1 - 2 *x)^(3/2))/64 + (98209*(1 - 2*x)^(5/2))/40 - (260055*(1 - 2*x)^(7/2))/448 + (1275*(1 - 2*x)^(9/2))/16 - (3375*(1 - 2*x)^(11/2))/704
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx=\frac {-118125 x^{6}-627375 x^{5}-1564350 x^{4}-2569643 x^{3}-3611453 x^{2}-8012926 x +8096086}{385 \sqrt {-2 x +1}} \] Input:
int((2+3*x)^3*(3+5*x)^3/(1-2*x)^(3/2),x)
Output:
( - 118125*x**6 - 627375*x**5 - 1564350*x**4 - 2569643*x**3 - 3611453*x**2 - 8012926*x + 8096086)/(385*sqrt( - 2*x + 1))