Integrand size = 24, antiderivative size = 105 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=\frac {3195731}{384 (1-2 x)^{3/2}}-\frac {9836211}{128 \sqrt {1-2 x}}-\frac {12973191}{128} \sqrt {1-2 x}+\frac {9504551}{384} (1-2 x)^{3/2}-\frac {4177401}{640} (1-2 x)^{5/2}+\frac {1101465}{896} (1-2 x)^{7/2}-\frac {17925}{128} (1-2 x)^{9/2}+\frac {10125 (1-2 x)^{11/2}}{1408} \]
3195731/384/(1-2*x)^(3/2)+9504551/384*(1-2*x)^(3/2)-4177401/640*(1-2*x)^(5 /2)+1101465/896*(1-2*x)^(7/2)-17925/128*(1-2*x)^(9/2)+10125/1408*(1-2*x)^( 11/2)-9836211/128/(1-2*x)^(1/2)-12973191/128*(1-2*x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=-\frac {173891632-522173856 x+258342648 x^2+77493296 x^3+41201532 x^4+19961775 x^5+6630750 x^6+1063125 x^7}{1155 (1-2 x)^{3/2}} \]
-1/1155*(173891632 - 522173856*x + 258342648*x^2 + 77493296*x^3 + 41201532 *x^4 + 19961775*x^5 + 6630750*x^6 + 1063125*x^7)/(1 - 2*x)^(3/2)
Time = 0.19 (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.
\(\displaystyle \int \frac {(3 x+2)^4 (5 x+3)^3}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {10125}{128} (1-2 x)^{9/2}+\frac {161325}{128} (1-2 x)^{7/2}-\frac {1101465}{128} (1-2 x)^{5/2}+\frac {4177401}{128} (1-2 x)^{3/2}-\frac {9504551}{128} \sqrt {1-2 x}+\frac {12973191}{128 \sqrt {1-2 x}}-\frac {9836211}{128 (1-2 x)^{3/2}}+\frac {3195731}{128 (1-2 x)^{5/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {10125 (1-2 x)^{11/2}}{1408}-\frac {17925}{128} (1-2 x)^{9/2}+\frac {1101465}{896} (1-2 x)^{7/2}-\frac {4177401}{640} (1-2 x)^{5/2}+\frac {9504551}{384} (1-2 x)^{3/2}-\frac {12973191}{128} \sqrt {1-2 x}-\frac {9836211}{128 \sqrt {1-2 x}}+\frac {3195731}{384 (1-2 x)^{3/2}}\) |
3195731/(384*(1 - 2*x)^(3/2)) - 9836211/(128*Sqrt[1 - 2*x]) - (12973191*Sq rt[1 - 2*x])/128 + (9504551*(1 - 2*x)^(3/2))/384 - (4177401*(1 - 2*x)^(5/2 ))/640 + (1101465*(1 - 2*x)^(7/2))/896 - (17925*(1 - 2*x)^(9/2))/128 + (10 125*(1 - 2*x)^(11/2))/1408
3.22.58.3.1 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 = 1.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {1063125 x^{7}+6630750 x^{6}+19961775 x^{5}+41201532 x^{4}+77493296 x^{3}+258342648 x^{2}-522173856 x +173891632}{1155 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
pseudoelliptic | \(\frac {-1063125 x^{7}-6630750 x^{6}-19961775 x^{5}-41201532 x^{4}-77493296 x^{3}-258342648 x^{2}+522173856 x -173891632}{1155 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(45\) |
trager | \(-\frac {\left (1063125 x^{7}+6630750 x^{6}+19961775 x^{5}+41201532 x^{4}+77493296 x^{3}+258342648 x^{2}-522173856 x +173891632\right ) \sqrt {1-2 x}}{1155 \left (-1+2 x \right )^{2}}\) | \(52\) |
risch | \(\frac {1063125 x^{7}+6630750 x^{6}+19961775 x^{5}+41201532 x^{4}+77493296 x^{3}+258342648 x^{2}-522173856 x +173891632}{1155 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(52\) |
derivativedivides | \(\frac {3195731}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {9504551 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {4177401 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {1101465 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {17925 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {10125 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {9836211}{128 \sqrt {1-2 x}}-\frac {12973191 \sqrt {1-2 x}}{128}\) | \(74\) |
default | \(\frac {3195731}{384 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {9504551 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {4177401 \left (1-2 x \right )^{\frac {5}{2}}}{640}+\frac {1101465 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {17925 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {10125 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {9836211}{128 \sqrt {1-2 x}}-\frac {12973191 \sqrt {1-2 x}}{128}\) | \(74\) |
meijerg | \(-\frac {288 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {1584 \sqrt {\pi }-\frac {198 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {3732 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {117184 \sqrt {\pi }}{3}-\frac {1831 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{6 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {30649 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}+\frac {\frac {230760 \sqrt {\pi }}{7}-\frac {28845 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {15075 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}+\frac {\frac {18000 \sqrt {\pi }}{11}-\frac {1125 \sqrt {\pi }\, \left (18432 x^{7}+14336 x^{6}+12288 x^{5}+12288 x^{4}+16384 x^{3}+49152 x^{2}-98304 x +32768\right )}{22528 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\) | \(324\) |
-1/1155/(1-2*x)^(3/2)*(1063125*x^7+6630750*x^6+19961775*x^5+41201532*x^4+7 7493296*x^3+258342648*x^2-522173856*x+173891632)
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=-\frac {{\left (1063125 \, x^{7} + 6630750 \, x^{6} + 19961775 \, x^{5} + 41201532 \, x^{4} + 77493296 \, x^{3} + 258342648 \, x^{2} - 522173856 \, x + 173891632\right )} \sqrt {-2 \, x + 1}}{1155 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/1155*(1063125*x^7 + 6630750*x^6 + 19961775*x^5 + 41201532*x^4 + 7749329 6*x^3 + 258342648*x^2 - 522173856*x + 173891632)*sqrt(-2*x + 1)/(4*x^2 - 4 *x + 1)
Time = 1.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=\frac {10125 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {17925 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {1101465 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {4177401 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} + \frac {9504551 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} - \frac {12973191 \sqrt {1 - 2 x}}{128} - \frac {9836211}{128 \sqrt {1 - 2 x}} + \frac {3195731}{384 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
10125*(1 - 2*x)**(11/2)/1408 - 17925*(1 - 2*x)**(9/2)/128 + 1101465*(1 - 2 *x)**(7/2)/896 - 4177401*(1 - 2*x)**(5/2)/640 + 9504551*(1 - 2*x)**(3/2)/3 84 - 12973191*sqrt(1 - 2*x)/128 - 9836211/(128*sqrt(1 - 2*x)) + 3195731/(3 84*(1 - 2*x)**(3/2))
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=\frac {10125}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {17925}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1101465}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4177401}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {9504551}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {12973191}{128} \, \sqrt {-2 \, x + 1} + \frac {41503 \, {\left (711 \, x - 317\right )}}{192 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
10125/1408*(-2*x + 1)^(11/2) - 17925/128*(-2*x + 1)^(9/2) + 1101465/896*(- 2*x + 1)^(7/2) - 4177401/640*(-2*x + 1)^(5/2) + 9504551/384*(-2*x + 1)^(3/ 2) - 12973191/128*sqrt(-2*x + 1) + 41503/192*(711*x - 317)/(-2*x + 1)^(3/2 )
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=-\frac {10125}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {17925}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1101465}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4177401}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {9504551}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {12973191}{128} \, \sqrt {-2 \, x + 1} - \frac {41503 \, {\left (711 \, x - 317\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
-10125/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 17925/128*(2*x - 1)^4*sqrt(-2*x + 1) - 1101465/896*(2*x - 1)^3*sqrt(-2*x + 1) - 4177401/640*(2*x - 1)^2*sqr t(-2*x + 1) + 9504551/384*(-2*x + 1)^(3/2) - 12973191/128*sqrt(-2*x + 1) - 41503/192*(711*x - 317)/((2*x - 1)*sqrt(-2*x + 1))
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^4 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx=\frac {\frac {9836211\,x}{64}-\frac {13156451}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {12973191\,\sqrt {1-2\,x}}{128}+\frac {9504551\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {4177401\,{\left (1-2\,x\right )}^{5/2}}{640}+\frac {1101465\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {17925\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {10125\,{\left (1-2\,x\right )}^{11/2}}{1408} \]