Integrand size = 28, antiderivative size = 280 \[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {49213766267 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4560806250}-\frac {1395767753 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{202702500}-\frac {14917646 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{6081075}-\frac {23321 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{26730}-\frac {3481 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{19305}-\frac {89 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {836091184171 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{1954631250 \sqrt {35}}+\frac {48128081531 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{3909262500 \sqrt {35}} \] Output:
49213766267/4560806250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-139576775 3/202702500*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-14917646/6081075*(1- 2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-23321/26730*(1-2*x)^(3/2)*(2+3*x)^( 5/2)*(3+5*x)^(1/2)-3481/19305*(1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2)-89 /2925*(1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2)+2/75*(1-2*x)^(3/2)*(2+3*x) ^(5/2)*(3+5*x)^(7/2)-836091184171/68412093750*EllipticE(1/11*55^(1/2)*(1-2 *x)^(1/2),1/35*1155^(1/2))*35^(1/2)+48128081531/136824187500*EllipticF(1/1 1*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 7.95 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.43 \[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {1672182368342 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (44426819351-177853891770 x-522917547750 x^2-227285730000 x^3+888419542500 x^4+1316318850000 x^5+547296750000 x^6\right )+344496363869 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{136824187500} \] Input:
Integrate[(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2),x]
Output:
((1672182368342*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5*(3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(44426819351 - 177853891770 *x - 522917547750*x^2 - 227285730000*x^3 + 888419542500*x^4 + 131631885000 0*x^5 + 547296750000*x^6) + (344496363869*I)*Sqrt[33]*EllipticF[I*ArcSinh[ Sqrt[9 + 15*x]], -2/33]))/136824187500
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, 25, 171, 27, 171, 27, 171, 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.
| \(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac {2}{75} \int -\frac {1}{2} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2} (89 x+71)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \int \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2} (89 x+71)dx+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {2}{195} \int \frac {(4678-2503 x) (3 x+2)^{3/2} (5 x+3)^{5/2}}{2 \sqrt {1-2 x}}dx+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \int \frac {(4678-2503 x) (3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}dx+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac {1}{55} \int -\frac {3 \sqrt {3 x+2} (5 x+3)^{5/2} (399442 x+272135)}{2 \sqrt {1-2 x}}dx\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \int \frac {\sqrt {3 x+2} (5 x+3)^{5/2} (399442 x+272135)}{\sqrt {1-2 x}}dx+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (-\frac {1}{45} \int -\frac {(5 x+3)^{5/2} (57509209 x+37873457)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \int \frac {(5 x+3)^{5/2} (57509209 x+37873457)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (-\frac {1}{21} \int -\frac {5 (5 x+3)^{3/2} (2280795702 x+1494997681)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \int \frac {(5 x+3)^{3/2} (2280795702 x+1494997681)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (50299451003 x+32688545874)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (50299451003 x+32688545874)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {3344364736684 x+2117277634217}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {50299451003}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {3344364736684 x+2117277634217}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {50299451003}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {553293961033}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3344364736684}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {50299451003}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {553293961033}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {3344364736684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {50299451003}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{75} \left (\frac {1}{195} \left (\frac {3}{110} \left (\frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {100598902006}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {3344364736684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {50299451003}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760265234}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {57509209}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {399442}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )+\frac {2503}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {178}{195} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\right )+\frac {2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
Input:
Int[(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2),x]
Output:
(2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/75 + ((178*Sqrt[1 - 2* x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/195 + ((2503*Sqrt[1 - 2*x]*(2 + 3*x)^( 3/2)*(3 + 5*x)^(7/2))/55 + (3*((-399442*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5 *x)^(7/2))/45 + ((-57509209*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/2 1 + (5*((-760265234*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-50 299451003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-3344364736684* Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (1005989 02006*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18) /5))/42)/45))/110)/195)/75
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 = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (492567075000000 x^{9}+1562321722500000 x^{8}+1592905277250000 x^{7}+33511953825000 x^{6}+1659881883099 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3344364736684 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1050958443600000 x^{5}-633067124890500 x^{4}+67989068522100 x^{3}+162128981218890 x^{2}+22684068454890 x -7996827483180\right )}{273648375000 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(168\) |
| risch | \(\frac {\left (547296750000 x^{6}+1316318850000 x^{5}+888419542500 x^{4}-227285730000 x^{3}-522917547750 x^{2}-177853891770 x +44426819351\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{9121612500 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {2117277634217 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{1003377375000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {836091184171 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{250844343750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(271\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {1976154353 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{101351250}-\frac {44426819351 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9121612500}+\frac {2117277634217 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{383107725000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {836091184171 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{95776931250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {17877523 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{311850}+\frac {481028 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19305}-\frac {69639 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{715}-\frac {1876 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}-60 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(334\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/273648375000*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(492567075000000 *x^9+1562321722500000*x^8+1592905277250000*x^7+33511953825000*x^6+16598818 83099*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28 +42*x)^(1/2),1/2*70^(1/2))-3344364736684*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1 /2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-105095844360 0000*x^5-633067124890500*x^4+67989068522100*x^3+162128981218890*x^2+226840 68454890*x-7996827483180)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.28 \[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=-\frac {1}{9121612500} \, {\left (547296750000 \, x^{6} + 1316318850000 \, x^{5} + 888419542500 \, x^{4} - 227285730000 \, x^{3} - 522917547750 \, x^{2} - 177853891770 \, x + 44426819351\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {56817299067899}{12314176875000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {836091184171}{68412093750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/9121612500*(547296750000*x^6 + 1316318850000*x^5 + 888419542500*x^4 - 2 27285730000*x^3 - 522917547750*x^2 - 177853891770*x + 44426819351)*sqrt(5* x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 56817299067899/12314176875000*sqrt(- 30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 836091184171/6 8412093750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPIn verse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)**(5/2)*(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(3/2), x)
\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(3/2), x)
Timed out. \[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \] Input:
int((1 - 2*x)^(3/2)*(3*x + 2)^(5/2)*(5*x + 3)^(5/2),x)
Output:
int((1 - 2*x)^(3/2)*(3*x + 2)^(5/2)*(5*x + 3)^(5/2), x)
\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=-60 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{6}-\frac {1876 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{13}-\frac {69639 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{715}+\frac {481028 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{19305}+\frac {17877523 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{311850}+\frac {1976154353 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{101351250}-\frac {19955549729 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1554052500}+\frac {836091184171 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{2331078750}-\frac {49714125239 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{345345000} \] Input:
int((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x)
Output:
( - 559458900000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**6 - 13455 70380000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 - 908162199000* sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 + 232336524000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 534537937700*sqrt(3*x + 2)*sqrt (5*x + 3)*sqrt( - 2*x + 1)*x**2 + 181806200476*sqrt(3*x + 2)*sqrt(5*x + 3) *sqrt( - 2*x + 1)*x - 119733298374*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 3344364736684*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x* *2)/(30*x**3 + 23*x**2 - 7*x - 6),x) - 1342281381453*int((sqrt(3*x + 2)*sq rt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/9324315000