Integrand size = 22, antiderivative size = 92 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {184877}{64 \sqrt {1-2 x}}+\frac {60025}{8} \sqrt {1-2 x}-\frac {173215}{64} (1-2 x)^{3/2}+\frac {7497}{8} (1-2 x)^{5/2}-\frac {13905}{64} (1-2 x)^{7/2}+\frac {117}{4} (1-2 x)^{9/2}-\frac {1215}{704} (1-2 x)^{11/2} \]
-173215/64*(1-2*x)^(3/2)+7497/8*(1-2*x)^(5/2)-13905/64*(1-2*x)^(7/2)+117/4 *(1-2*x)^(9/2)-1215/704*(1-2*x)^(11/2)+184877/64/(1-2*x)^(1/2)+60025/8*(1- 2*x)^(1/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {92760-91704 x-41012 x^2-28692 x^3-17055 x^4-6651 x^5-1215 x^6}{11 \sqrt {1-2 x}} \]
(92760 - 91704*x - 41012*x^2 - 28692*x^3 - 17055*x^4 - 6651*x^5 - 1215*x^6 )/(11*Sqrt[1 - 2*x])
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {86, 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.
\(\displaystyle \int \frac {(3 x+2)^5 (5 x+3)}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {1215}{64} (1-2 x)^{9/2}-\frac {1053}{4} (1-2 x)^{7/2}+\frac {97335}{64} (1-2 x)^{5/2}-\frac {37485}{8} (1-2 x)^{3/2}+\frac {519645}{64} \sqrt {1-2 x}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {184877}{64 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1215}{704} (1-2 x)^{11/2}+\frac {117}{4} (1-2 x)^{9/2}-\frac {13905}{64} (1-2 x)^{7/2}+\frac {7497}{8} (1-2 x)^{5/2}-\frac {173215}{64} (1-2 x)^{3/2}+\frac {60025}{8} \sqrt {1-2 x}+\frac {184877}{64 \sqrt {1-2 x}}\) |
184877/(64*Sqrt[1 - 2*x]) + (60025*Sqrt[1 - 2*x])/8 - (173215*(1 - 2*x)^(3 /2))/64 + (7497*(1 - 2*x)^(5/2))/8 - (13905*(1 - 2*x)^(7/2))/64 + (117*(1 - 2*x)^(9/2))/4 - (1215*(1 - 2*x)^(11/2))/704
3.21.71.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 1.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {1215 x^{6}+6651 x^{5}+17055 x^{4}+28692 x^{3}+41012 x^{2}+91704 x -92760}{11 \sqrt {1-2 x}}\) | \(40\) |
risch | \(-\frac {1215 x^{6}+6651 x^{5}+17055 x^{4}+28692 x^{3}+41012 x^{2}+91704 x -92760}{11 \sqrt {1-2 x}}\) | \(40\) |
pseudoelliptic | \(\frac {-1215 x^{6}-6651 x^{5}-17055 x^{4}-28692 x^{3}-41012 x^{2}-91704 x +92760}{11 \sqrt {1-2 x}}\) | \(40\) |
trager | \(\frac {\left (1215 x^{6}+6651 x^{5}+17055 x^{4}+28692 x^{3}+41012 x^{2}+91704 x -92760\right ) \sqrt {1-2 x}}{-11+22 x}\) | \(47\) |
derivativedivides | \(-\frac {173215 \left (1-2 x \right )^{\frac {3}{2}}}{64}+\frac {7497 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{4}-\frac {1215 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {184877}{64 \sqrt {1-2 x}}+\frac {60025 \sqrt {1-2 x}}{8}\) | \(65\) |
default | \(-\frac {173215 \left (1-2 x \right )^{\frac {3}{2}}}{64}+\frac {7497 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {13905 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{4}-\frac {1215 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {184877}{64 \sqrt {1-2 x}}+\frac {60025 \sqrt {1-2 x}}{8}\) | \(65\) |
meijerg | \(-\frac {96 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-880 \sqrt {\pi }+\frac {110 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {840 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-2736 \sqrt {\pi }+\frac {171 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3915 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{8 \sqrt {\pi }}+\frac {-\frac {4248 \sqrt {\pi }}{7}+\frac {531 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {1215 \left (\frac {1024 \sqrt {\pi }}{231}-\frac {\sqrt {\pi }\, \left (-2688 x^{6}-1792 x^{5}-1280 x^{4}-1024 x^{3}-1024 x^{2}-2048 x +2048\right )}{462 \sqrt {1-2 x}}\right )}{64 \sqrt {\pi }}\) | \(266\) |
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {{\left (1215 \, x^{6} + 6651 \, x^{5} + 17055 \, x^{4} + 28692 \, x^{3} + 41012 \, x^{2} + 91704 \, x - 92760\right )} \sqrt {-2 \, x + 1}}{11 \, {\left (2 \, x - 1\right )}} \]
1/11*(1215*x^6 + 6651*x^5 + 17055*x^4 + 28692*x^3 + 41012*x^2 + 91704*x - 92760)*sqrt(-2*x + 1)/(2*x - 1)
Time = 1.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=- \frac {1215 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {117 \left (1 - 2 x\right )^{\frac {9}{2}}}{4} - \frac {13905 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {7497 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {173215 \left (1 - 2 x\right )^{\frac {3}{2}}}{64} + \frac {60025 \sqrt {1 - 2 x}}{8} + \frac {184877}{64 \sqrt {1 - 2 x}} \]
-1215*(1 - 2*x)**(11/2)/704 + 117*(1 - 2*x)**(9/2)/4 - 13905*(1 - 2*x)**(7 /2)/64 + 7497*(1 - 2*x)**(5/2)/8 - 173215*(1 - 2*x)**(3/2)/64 + 60025*sqrt (1 - 2*x)/8 + 184877/(64*sqrt(1 - 2*x))
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=-\frac {1215}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {117}{4} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {13905}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {7497}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {173215}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {60025}{8} \, \sqrt {-2 \, x + 1} + \frac {184877}{64 \, \sqrt {-2 \, x + 1}} \]
-1215/704*(-2*x + 1)^(11/2) + 117/4*(-2*x + 1)^(9/2) - 13905/64*(-2*x + 1) ^(7/2) + 7497/8*(-2*x + 1)^(5/2) - 173215/64*(-2*x + 1)^(3/2) + 60025/8*sq rt(-2*x + 1) + 184877/64/sqrt(-2*x + 1)
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {1215}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {117}{4} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {13905}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {7497}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {173215}{64} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {60025}{8} \, \sqrt {-2 \, x + 1} + \frac {184877}{64 \, \sqrt {-2 \, x + 1}} \]
1215/704*(2*x - 1)^5*sqrt(-2*x + 1) + 117/4*(2*x - 1)^4*sqrt(-2*x + 1) + 1 3905/64*(2*x - 1)^3*sqrt(-2*x + 1) + 7497/8*(2*x - 1)^2*sqrt(-2*x + 1) - 1 73215/64*(-2*x + 1)^(3/2) + 60025/8*sqrt(-2*x + 1) + 184877/64/sqrt(-2*x + 1)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^5 (3+5 x)}{(1-2 x)^{3/2}} \, dx=\frac {184877}{64\,\sqrt {1-2\,x}}+\frac {60025\,\sqrt {1-2\,x}}{8}-\frac {173215\,{\left (1-2\,x\right )}^{3/2}}{64}+\frac {7497\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {13905\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {117\,{\left (1-2\,x\right )}^{9/2}}{4}-\frac {1215\,{\left (1-2\,x\right )}^{11/2}}{704} \]