Integrand size = 28, antiderivative size = 280 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{53093313 \sqrt {33}} \]
-2/39*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2)+230/1287*(1-2*x)^(3/2)*(3 +5*x)^(3/2)/(2+3*x)^(11/2)-75041008472/1752079329*EllipticE(1/7*21^(1/2)*( 1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2257166048/1752079329*EllipticF(1/7 *21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1300/891*(3+5*x)^(3/2)*( 1-2*x)^(1/2)/(2+3*x)^(9/2)-3347620/1702701*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+ 3*x)^(7/2)+23210828/11918907*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+107 9936248/83432349*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+75041008472/584 026443*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 9.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.41 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2532151719515+22577209892436 x+83893544414217 x^2+166295375376786 x^3+185457331738206 x^4+110328276131100 x^5+27352447588044 x^6\right )}{4 (2+3 x)^{13/2}}+i \sqrt {33} \left (9380126059 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-9662271815 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1752079329} \]
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2532151719515 + 22577209892436*x + 838 93544414217*x^2 + 166295375376786*x^3 + 185457331738206*x^4 + 110328276131 100*x^5 + 27352447588044*x^6))/(4*(2 + 3*x)^(13/2)) + I*Sqrt[33]*(93801260 59*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 9662271815*EllipticF[I*Ar cSinh[Sqrt[9 + 15*x]], -2/33])))/1752079329
Time = 0.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {108, 27, 167, 27, 167, 27, 167, 27, 169, 27, 169, 27, 169, 27, 176, 123, 129}
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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^{15/2}} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {2}{39} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)^{13/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{3/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{(3 x+2)^{13/2}}dx-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \int \frac {3 (193-133 x) \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^{11/2}}dx-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \int \frac {(193-133 x) \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^{11/2}}dx-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}-\frac {2}{27} \int -\frac {(29223-39856 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{9/2}}dx\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \int \frac {(29223-39856 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {2}{147} \int \frac {1003789-1395670 x}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \int \frac {1003789-1395670 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {2}{35} \int \frac {3 (25654987-29013535 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \int \frac {25654987-29013535 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {2226761813-1349920310 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {2226761813-1349920310 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (9380126059 x+5938435967)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {18760252118 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {9380126059 x+5938435967}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {18760252118 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1551801658}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9380126059}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {18760252118 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1551801658}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9380126059}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {18760252118 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle -\frac {5}{39} \left (-\frac {2}{11} \left (\frac {1}{27} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {282145756}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {9380126059}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {18760252118 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {269984062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {11605414 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {334762 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {1690 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(39*(2 + 3*x)^(13/2)) - (5*((-46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) - (2*((1690*Sqrt[1 - 2* x]*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) + ((-334762*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) + ((11605414*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3 5*(2 + 3*x)^(5/2)) + (6*((269984062*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + ((18760252118*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (10*((-9380126059*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (282145756*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5))/7)/21))/35)/147)/27))/11))/39
3.28.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.25
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {7616 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{6}}+\frac {18134 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{5}}+\frac {390100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137918781 \left (\frac {2}{3}+x \right )^{4}}+\frac {23210828 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321810489 \left (\frac {2}{3}+x \right )^{3}}+\frac {1079936248 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{750891141 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {750410084720}{584026443} x^{2}-\frac {75041008472}{584026443} x +\frac {75041008472}{194675481}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {95014975472 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {150082016944 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2302911 \left (\frac {2}{3}+x \right )^{7}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(350\) |
default | \(\frac {2 \left (27352447588044 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-26565174063252 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+109409790352176 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-106260696253008 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+182349650586960 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-177101160421680 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+162088578299520 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-157423253708160 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+81044289149760 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-78711626854080 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+820573427641320 x^{8}+21611810439936 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-20989767161088 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3391905626697132 x^{7}+2401312271104 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2332196351232 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+5648532752247084 x^{6}+4552278771338298 x^{5}+1346576472913014 x^{4}-567661448375343 x^{3}-611345718465195 x^{2}-195598433873379 x -22789365475635\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{1752079329 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\) | \(694\) |
-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5* x)*(2+3*x))^(1/2)*(-7616/8444007*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6+18 134/8444007*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5+390100/137918781*(-30*x ^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+23210828/321810489*(-30*x^3-23*x^2+7*x+6) ^(1/2)/(2/3+x)^3+1079936248/750891141*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x) ^2+75041008472/1752079329*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+ 95014975472/12264555303*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(- 30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+150082 016944/12264555303*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^ 3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*E llipticF((10+15*x)^(1/2),1/35*70^(1/2)))+98/2302911*(-30*x^3-23*x^2+7*x+6) ^(1/2)/(2/3+x)^7)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\frac {2 \, {\left (135 \, {\left (27352447588044 \, x^{6} + 110328276131100 \, x^{5} + 185457331738206 \, x^{4} + 166295375376786 \, x^{3} + 83893544414217 \, x^{2} + 22577209892436 \, x + 2532151719515\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 637432675346 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1688422690620 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{78843569805 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
2/78843569805*(135*(27352447588044*x^6 + 110328276131100*x^5 + 18545733173 8206*x^4 + 166295375376786*x^3 + 83893544414217*x^2 + 22577209892436*x + 2 532151719515)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 637432675346*sq rt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x ^2 + 1344*x + 128)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1688422690620*sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 1 5120*x^3 + 6048*x^2 + 1344*x + 128)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2187*x^7 + 10206 *x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]