Integrand size = 28, antiderivative size = 249 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=-\frac {69808931 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{168918750}-\frac {445024 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{9384375}+\frac {32717 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1126125}+\frac {34 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2475}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}{2145}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {1163388067 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{38390625 \sqrt {33}}-\frac {69808931 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{76781250 \sqrt {33}} \]
62/2145*(1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2)+2/65*(1-2*x)^(5/2)*(2+3* x)^(5/2)*(3+5*x)^(3/2)-1163388067/1266890625*EllipticE(1/7*21^(1/2)*(1-2*x )^(1/2),1/33*1155^(1/2))*33^(1/2)-69808931/2533781250*EllipticF(1/7*21^(1/ 2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+32717/1126125*(2+3*x)^(3/2)*(3+ 5*x)^(3/2)*(1-2*x)^(1/2)+34/2475*(2+3*x)^(5/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2) -445024/9384375*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-69808931/1689187 50*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (84411073+380959290 x-309143250 x^2-936022500 x^3+433755000 x^4+935550000 x^5\right )+2326776134 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-2396585065 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{2533781250} \]
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(84411073 + 380959290*x - 30 9143250*x^2 - 936022500*x^3 + 433755000*x^4 + 935550000*x^5) + (2326776134 *I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (2396585065*I)* Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/2533781250
Time = 0.33 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^{5/2} \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{3/2}-\frac {2}{65} \int -\frac {5}{2} (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt {5 x+3} (31 x+23)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt {5 x+3} (31 x+23)dx+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{165} \int \frac {3}{2} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3} (663 x+862)dx+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \int \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3} (663 x+862)dx+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {2}{135} \int \frac {3 (24867-32717 x) (3 x+2)^{3/2} \sqrt {5 x+3}}{2 \sqrt {1-2 x}}dx+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \int \frac {(24867-32717 x) (3 x+2)^{3/2} \sqrt {5 x+3}}{\sqrt {1-2 x}}dx+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac {1}{35} \int -\frac {3 \sqrt {3 x+2} \sqrt {5 x+3} (890048 x+669705)}{2 \sqrt {1-2 x}}dx\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \int \frac {\sqrt {3 x+2} \sqrt {5 x+3} (890048 x+669705)}{\sqrt {1-2 x}}dx+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (-\frac {1}{25} \int -\frac {\sqrt {5 x+3} (69808931 x+45500898)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \int \frac {\sqrt {5 x+3} (69808931 x+45500898)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (-\frac {1}{9} \int -\frac {4653552268 x+2945711009}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {69808931}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \int \frac {4653552268 x+2945711009}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {69808931}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (\frac {767898241}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4653552268}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {69808931}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (\frac {767898241}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4653552268}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {69808931}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{55} \left (\frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (-\frac {139617862}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4653552268}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {69808931}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {890048}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {32717}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {442}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {62}{165} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/65 + ((62*(1 - 2*x)^(3 /2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/165 + ((442*Sqrt[1 - 2*x]*(2 + 3*x)^( 5/2)*(3 + 5*x)^(3/2))/45 + ((32717*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x) ^(3/2))/35 + (3*((-890048*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/25 + ((-69808931*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-4653552268 *Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (139617 862*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18)/2 5))/70)/45)/55)/13
3.28.54.3.1 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 = 1.40 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-420997500000 x^{8}-517954500000 x^{7}+2259637347 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2326776134 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+369797400000 x^{6}+591786000000 x^{5}-124021671750 x^{4}-286118004150 x^{3}-16943987235 x^{2}+43149498765 x +7596996570\right )}{2533781250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(165\) |
| risch | \(-\frac {\left (935550000 x^{5}+433755000 x^{4}-936022500 x^{3}-309143250 x^{2}+380959290 x +84411073\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{168918750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {2945711009 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{18581062500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1163388067 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{4645265625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {4232881 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1876875}+\frac {84411073 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{168918750}+\frac {2945711009 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{17736468750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2326776134 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{8868234375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {3523 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1925}+\frac {72 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}+\frac {1836 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{715}-\frac {3962 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{715}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(294\) |
-1/2533781250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-420997500000*x^8 -517954500000*x^7+2259637347*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))-2326776134*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))+369797400000*x^6+591786000000*x^5-124021671750*x^4-28611 8004150*x^3-16943987235*x^2+43149498765*x+7596996570)/(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 = 74, normalized size of antiderivative = 0.30 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\frac {1}{168918750} \, {\left (935550000 \, x^{5} + 433755000 \, x^{4} - 936022500 \, x^{3} - 309143250 \, x^{2} + 380959290 \, x + 84411073\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {79041144323}{228040312500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1163388067}{1266890625} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/168918750*(935550000*x^5 + 433755000*x^4 - 936022500*x^3 - 309143250*x^2 + 380959290*x + 84411073)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 79 041144323/228040312500*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125 , x + 23/90) + 1163388067/1266890625*sqrt(-30)*weierstrassZeta(1159/675, 3 8998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\text {Timed out} \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int { \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int { \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3} \,d x \]