Integrand size = 24, antiderivative size = 92 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {290521}{192 (1-2 x)^{3/2}}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951}{64} \sqrt {1-2 x}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {225}{64} (1-2 x)^{9/2} \]
290521/192/(1-2*x)^(3/2)+40453/16*(1-2*x)^(3/2)-159111/320*(1-2*x)^(5/2)+1 3905/224*(1-2*x)^(7/2)-225/64*(1-2*x)^(9/2)-381073/32/(1-2*x)^(1/2)-832951 /64*(1-2*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {2238664-6731112 x+3294996 x^2+915492 x^3+402489 x^4+137700 x^5+23625 x^6}{105 (1-2 x)^{3/2}} \]
-1/105*(2238664 - 6731112*x + 3294996*x^2 + 915492*x^3 + 402489*x^4 + 1377 00*x^5 + 23625*x^6)/(1 - 2*x)^(3/2)
Time = 0.19 (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.
\(\displaystyle \int \frac {(3 x+2)^4 (5 x+3)^2}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {2025}{64} (1-2 x)^{7/2}-\frac {13905}{32} (1-2 x)^{5/2}+\frac {159111}{64} (1-2 x)^{3/2}-\frac {121359}{16} \sqrt {1-2 x}+\frac {832951}{64 \sqrt {1-2 x}}-\frac {381073}{32 (1-2 x)^{3/2}}+\frac {290521}{64 (1-2 x)^{5/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {225}{64} (1-2 x)^{9/2}+\frac {13905}{224} (1-2 x)^{7/2}-\frac {159111}{320} (1-2 x)^{5/2}+\frac {40453}{16} (1-2 x)^{3/2}-\frac {832951}{64} \sqrt {1-2 x}-\frac {381073}{32 \sqrt {1-2 x}}+\frac {290521}{192 (1-2 x)^{3/2}}\) |
290521/(192*(1 - 2*x)^(3/2)) - 381073/(32*Sqrt[1 - 2*x]) - (832951*Sqrt[1 - 2*x])/64 + (40453*(1 - 2*x)^(3/2))/16 - (159111*(1 - 2*x)^(5/2))/320 + ( 13905*(1 - 2*x)^(7/2))/224 - (225*(1 - 2*x)^(9/2))/64
3.22.47.3.1 Defintions of rubi rules used
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 = 1.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664}{105 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(40\) |
pseudoelliptic | \(\frac {-23625 x^{6}-137700 x^{5}-402489 x^{4}-915492 x^{3}-3294996 x^{2}+6731112 x -2238664}{105 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(40\) |
trager | \(-\frac {\left (23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664\right ) \sqrt {1-2 x}}{105 \left (-1+2 x \right )^{2}}\) | \(47\) |
risch | \(\frac {23625 x^{6}+137700 x^{5}+402489 x^{4}+915492 x^{3}+3294996 x^{2}-6731112 x +2238664}{105 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(47\) |
derivativedivides | \(\frac {290521}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {40453 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {159111 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {225 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951 \sqrt {1-2 x}}{64}\) | \(65\) |
default | \(\frac {290521}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {40453 \left (1-2 x \right )^{\frac {3}{2}}}{16}-\frac {159111 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{224}-\frac {225 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {381073}{32 \sqrt {1-2 x}}-\frac {832951 \sqrt {1-2 x}}{64}\) | \(65\) |
meijerg | \(-\frac {96 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {448 \sqrt {\pi }-\frac {56 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {2612 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {7216 \sqrt {\pi }-\frac {451 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {4203 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {20880 \sqrt {\pi }}{7}-\frac {1305 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{448 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {675 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}\) | \(266\) |
-1/105/(1-2*x)^(3/2)*(23625*x^6+137700*x^5+402489*x^4+915492*x^3+3294996*x ^2-6731112*x+2238664)
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {{\left (23625 \, x^{6} + 137700 \, x^{5} + 402489 \, x^{4} + 915492 \, x^{3} + 3294996 \, x^{2} - 6731112 \, x + 2238664\right )} \sqrt {-2 \, x + 1}}{105 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/105*(23625*x^6 + 137700*x^5 + 402489*x^4 + 915492*x^3 + 3294996*x^2 - 6 731112*x + 2238664)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)
Time = 1.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=- \frac {225 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {13905 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} - \frac {159111 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} + \frac {40453 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} - \frac {832951 \sqrt {1 - 2 x}}{64} - \frac {381073}{32 \sqrt {1 - 2 x}} + \frac {290521}{192 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
-225*(1 - 2*x)**(9/2)/64 + 13905*(1 - 2*x)**(7/2)/224 - 159111*(1 - 2*x)** (5/2)/320 + 40453*(1 - 2*x)**(3/2)/16 - 832951*sqrt(1 - 2*x)/64 - 381073/( 32*sqrt(1 - 2*x)) + 290521/(192*(1 - 2*x)**(3/2))
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {225}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {13905}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {159111}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} + \frac {3773 \, {\left (1212 \, x - 529\right )}}{192 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
-225/64*(-2*x + 1)^(9/2) + 13905/224*(-2*x + 1)^(7/2) - 159111/320*(-2*x + 1)^(5/2) + 40453/16*(-2*x + 1)^(3/2) - 832951/64*sqrt(-2*x + 1) + 3773/19 2*(1212*x - 529)/(-2*x + 1)^(3/2)
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.96 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=-\frac {225}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {13905}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {159111}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {40453}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {832951}{64} \, \sqrt {-2 \, x + 1} - \frac {3773 \, {\left (1212 \, x - 529\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
-225/64*(2*x - 1)^4*sqrt(-2*x + 1) - 13905/224*(2*x - 1)^3*sqrt(-2*x + 1) - 159111/320*(2*x - 1)^2*sqrt(-2*x + 1) + 40453/16*(-2*x + 1)^(3/2) - 8329 51/64*sqrt(-2*x + 1) - 3773/192*(1212*x - 529)/((2*x - 1)*sqrt(-2*x + 1))
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^4 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx=\frac {\frac {381073\,x}{16}-\frac {1995917}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {832951\,\sqrt {1-2\,x}}{64}+\frac {40453\,{\left (1-2\,x\right )}^{3/2}}{16}-\frac {159111\,{\left (1-2\,x\right )}^{5/2}}{320}+\frac {13905\,{\left (1-2\,x\right )}^{7/2}}{224}-\frac {225\,{\left (1-2\,x\right )}^{9/2}}{64} \]