Integrand size = 28, antiderivative size = 280 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {1580201444291 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}}-\frac {23763809947 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6219281250 \sqrt {33}} \]
106/4875*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(7/2)+2/75*(1-2*x)^(5/2)*(2+3 *x)^(3/2)*(3+5*x)^(7/2)-1580201444291/410472562500*EllipticE(1/7*21^(1/2)* (1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-23763809947/205236281250*EllipticF (1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+8038/804375*(2+3*x)^ (3/2)*(3+5*x)^(7/2)*(1-2*x)^(1/2)-359748241/1520268750*(3+5*x)^(3/2)*(1-2* x)^(1/2)*(2+3*x)^(1/2)-26534891/760134375*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3 *x)^(1/2)+364267/36196875*(3+5*x)^(7/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-237638 09947/13682418750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.67 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.42 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (9093216326+157612390605 x+59959633500 x^2-487924998750 x^3-352885207500 x^4+579573225000 x^5+547296750000 x^6\right )+1580201444291 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1627729064185 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{410472562500} \]
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9093216326 + 157612390605*x + 59959633500*x^2 - 487924998750*x^3 - 352885207500*x^4 + 579573225000*x^ 5 + 547296750000*x^6) + (1580201444291*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqr t[9 + 15*x]], -2/33] - (1627729064185*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt [9 + 15*x]], -2/33])/410472562500
Time = 0.34 (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 = {112, 27, 171, 27, 171, 27, 171, 27, 171, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac {2}{75} \int -\frac {1}{2} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2} (159 x+113)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \int (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2} (159 x+113)dx+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {2}{195} \int \frac {3}{2} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2} (4019 x+4112)dx+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \int \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2} (4019 x+4112)dx+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {2}{165} \int \frac {(336865-364267 x) \sqrt {3 x+2} (5 x+3)^{5/2}}{2 \sqrt {1-2 x}}dx+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \int \frac {(336865-364267 x) \sqrt {3 x+2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}dx+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}-\frac {1}{45} \int -\frac {(5 x+3)^{5/2} (53069782 x+36229811)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \int \frac {(5 x+3)^{5/2} (53069782 x+36229811)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (-\frac {1}{21} \int -\frac {5 (5 x+3)^{3/2} (1079244723 x+705923594)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \int \frac {(5 x+3)^{3/2} (1079244723 x+705923594)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (47527619894 x+30890910327)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (47527619894 x+30890910327)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {1580201444291 x+1000401248458}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {47527619894}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {1580201444291 x+1000401248458}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {47527619894}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {261401909417}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1580201444291}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {47527619894}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {261401909417}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1580201444291}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {47527619894}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{65} \left (\frac {1}{165} \left (\frac {1}{90} \left (\frac {5}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {47527619894}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1580201444291}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {47527619894}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {359748241}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {53069782}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {364267}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {8038}{165} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {106}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}\) |
(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/75 + ((106*(1 - 2*x)^( 3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/65 + ((8038*Sqrt[1 - 2*x]*(2 + 3*x)^ (3/2)*(3 + 5*x)^(7/2))/165 + ((364267*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x )^(7/2))/45 + ((-53069782*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/21 + (5*((-359748241*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-4752 7619894*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-1580201444291*Sq rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (475276198 94*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9)/10) )/21)/90)/165)/65)/75
3.28.75.3.1 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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 = 1.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-492567075000000 x^{9}-899250660000000 x^{8}+32623479000000 x^{7}+1534715974803 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1580201444291 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+902847084300000 x^{6}+312921865912500 x^{5}-349206885747000 x^{4}-192171420950850 x^{3}+37617016792110 x^{2}+30279805737360 x +1636778938680\right )}{410472562500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(170\) |
| risch | \(-\frac {\left (547296750000 x^{6}+579573225000 x^{5}-352885207500 x^{4}-487924998750 x^{3}+59959633500 x^{2}+157612390605 x +9093216326\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{13682418750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {500200624229 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{752533031250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1580201444291 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{1505066062500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {3502497569 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{304053750}+\frac {4546608163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6841209375}+\frac {500200624229 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{718326984375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1580201444291 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1436653968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2049902 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{467775}+40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}+\frac {1652 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{39}-\frac {55322 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{2145}-\frac {2065291 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{57915}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(328\) |
-1/410472562500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-49256707500000 0*x^9-899250660000000*x^8+32623479000000*x^7+1534715974803*5^(1/2)*(2+3*x) ^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35 *70^(1/2))-1580201444291*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5 *x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+902847084300000*x^6+312 921865912500*x^5-349206885747000*x^4-192171420950850*x^3+37617016792110*x^ 2+30279805737360*x+1636778938680)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.28 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {1}{13682418750} \, {\left (547296750000 \, x^{6} + 579573225000 \, x^{5} - 352885207500 \, x^{4} - 487924998750 \, x^{3} + 59959633500 \, x^{2} + 157612390605 \, x + 9093216326\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {53691479142527}{36942530625000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1580201444291}{410472562500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/13682418750*(547296750000*x^6 + 579573225000*x^5 - 352885207500*x^4 - 48 7924998750*x^3 + 59959633500*x^2 + 157612390605*x + 9093216326)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 53691479142527/36942530625000*sqrt(-30) *weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1580201444291/410 472562500*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInv erse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\text {Timed out} \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]