Integrand size = 28, antiderivative size = 311 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=-\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {3+5 x}}{183968329545 \sqrt {2+3 x}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}-\frac {12641611554328 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}}-\frac {380220959152 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16724393595 \sqrt {33}} \]
-2/45*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2)+74/351*(1-2*x)^(3/2)*(3+5 *x)^(5/2)/(2+3*x)^(13/2)-12641611554328/551904988635*EllipticE(1/7*21^(1/2 )*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-380220959152/551904988635*Ellipt icF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1085156/729729*(3 +5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)+16636/11583*(3+5*x)^(5/2)*(1-2*x)^ (1/2)/(2+3*x)^(11/2)-112817764/107270163*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3* x)^(7/2)+3914701972/3754455705*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+1 81941877952/26281189935*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+12641611 554328/183968329545*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.38 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (853124799464729+8886579657279639 x+39676146370896231 x^2+98427465692862075 x^3+146528498784887100 x^4+130900492508039982 x^5+64974368463330312 x^6+13823602234657668 x^7\right )}{(2+3 x)^{15/2}}+4 i \sqrt {33} \left (1580201444291 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1627729064185 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{551904988635} \]
(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853124799464729 + 8886579657279639*x + 39676146370896231*x^2 + 98427465692862075*x^3 + 146528498784887100*x^4 + 130900492508039982*x^5 + 64974368463330312*x^6 + 13823602234657668*x^7))/( 2 + 3*x)^(15/2) + (4*I)*Sqrt[33]*(1580201444291*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1627729064185*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/ 33])))/551904988635
Time = 0.36 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {108, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{17/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{45} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^{15/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{9} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^{15/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \int \frac {(943-665 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^{13/2}}dx+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {2}{33} \int -\frac {(142409-193320 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^{11/2}}dx\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \int \frac {(142409-193320 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{11/2}}dx+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {2}{189} \int \frac {3 (4912047-6734150 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{9/2}}dx-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \int \frac {(4912047-6734150 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {2}{147} \int \frac {166462031-239644700 x}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \int \frac {166462031-239644700 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {2}{35} \int \frac {3 (4318659938-4893377465 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \int \frac {4318659938-4893377465 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {375115583137-227427347440 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {375115583137-227427347440 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (1580201444291 x+1000401248458)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3160402888582 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {1580201444291 x+1000401248458}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3160402888582 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {261401909417}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1580201444291}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {3160402888582 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {261401909417}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1580201444291}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3160402888582 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{39} \left (\frac {1}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {47527619894}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1580201444291}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3160402888582 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {45485469488 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {1957350986 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {56408882 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {542578 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {8318 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(45*(2 + 3*x)^(15/2)) + ((74*(1 - 2*x )^(3/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/2)) + (2*((8318*Sqrt[1 - 2*x]*( 3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + ((-542578*Sqrt[1 - 2*x]*(3 + 5*x)^ (3/2))/(63*(2 + 3*x)^(9/2)) + ((-56408882*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14 7*(2 + 3*x)^(7/2)) + ((1957350986*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3* x)^(5/2)) + (6*((45485469488*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3 /2)) + ((3160402888582*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (1 0*((-1580201444291*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5 - (47527619894*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5))/7)/21))/35)/147)/63)/33))/39)/9
3.28.85.3.1 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 = 1.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.20
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23914845 \left (\frac {2}{3}+x \right )^{8}}+\frac {16058 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{103630995 \left (\frac {2}{3}+x \right )^{7}}-\frac {641434 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{379980315 \left (\frac {2}{3}+x \right )^{6}}+\frac {1813814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{531972441 \left (\frac {2}{3}+x \right )^{5}}+\frac {1513936 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8688883203 \left (\frac {2}{3}+x \right )^{4}}+\frac {3914701972 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{101370304035 \left (\frac {2}{3}+x \right )^{3}}+\frac {181941877952 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236530709415 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {25283223108656}{36793665909} x^{2}-\frac {12641611554328}{183968329545} x +\frac {12641611554328}{61322776515}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {16006419975328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{3863334920445 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {25283223108656 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3863334920445 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(374\) |
| default | \(-\frac {2 \left (77419842517122564 x +785774579099136 \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 )-809063139476992 \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 )+139229433234128160 \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}-143355875026079520 \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}-2214305034568163712 x^{5}-4203787124900760138 x^{6}-3997525460519271384 x^{7}+500221362404680812 x^{3}-166810299141489255 x^{4}+304831834382285292 x^{2}-1990701860603882364 x^{8}-95570583350719680 \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}-414708067039730040 x^{9}+8250633080540928 \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}-8495162964508416 \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}+37127848862434176 \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}-38228233340287872 \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}+125306489910715344 \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}-129020287523471568 \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}+62653244955357672 \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}-64510143761735784 \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}+92819622156085440 \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}+13425695347576644 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-13823602234657668 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+7678123195182561\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{551904988635 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {15}{2}}}\) | \(789\) |
-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)*(-98/23914845*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^8+160 58/103630995*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^7-641434/379980315*(-30* x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6+1813814/531972441*(-30*x^3-23*x^2+7*x+6) ^(1/2)/(2/3+x)^5+1513936/8688883203*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4 +3914701972/101370304035*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+1819418779 52/236530709415*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+12641611554328/5519 04988635*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+16006419975328/38 63334920445*(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))+25283223108656/386 3334920445*(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*EllipticF ((10+15*x)^(1/2),1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\frac {2 \, {\left (135 \, {\left (13823602234657668 \, x^{7} + 64974368463330312 \, x^{6} + 130900492508039982 \, x^{5} + 146528498784887100 \, x^{4} + 98427465692862075 \, x^{3} + 39676146370896231 \, x^{2} + 8886579657279639 \, x + 853124799464729\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 107382958285054 \, \sqrt {-30} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 284436259972380 \, \sqrt {-30} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{24835724488575 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} \]
2/24835724488575*(135*(13823602234657668*x^7 + 64974368463330312*x^6 + 130 900492508039982*x^5 + 146528498784887100*x^4 + 98427465692862075*x^3 + 396 76146370896231*x^2 + 8886579657279639*x + 853124799464729)*sqrt(5*x + 3)*s qrt(3*x + 2)*sqrt(-2*x + 1) - 107382958285054*sqrt(-30)*(6561*x^8 + 34992* x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 28443625997 2380*sqrt(-30)*(6561*x^8 + 34992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256)*weierstrassZeta(1159/675, 38998/91 125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(6561*x^8 + 3 4992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 30 72*x + 256)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{17/2}} \,d x \]