Integrand size = 28, antiderivative size = 218 \[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {90397364 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}-\frac {5442127 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3543750 \sqrt {33}} \]
2/55*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2)-90397364/58471875*EllipticE (1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-5442127/116943750*El lipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+62/2475*(2+3* x)^(3/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)-40703/433125*(3+5*x)^(3/2)*(1-2*x)^(1 /2)*(2+3*x)^(1/2)-23/9625*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-544212 7/7796250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\frac {180794728 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-810641-27227430 x-17237250 x^2+43470000 x^3+42525000 x^4\right )+37247371 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{116943750} \]
((180794728*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]*(-810641 - 27227430*x - 1723725 0*x^2 + 43470000*x^3 + 42525000*x^4) + (37247371*I)*Sqrt[33]*EllipticF[I*A rcSinh[Sqrt[9 + 15*x]], -2/33]))/116943750
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {112, 27, 171, 27, 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.
| \(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {2}{55} \int -\frac {3}{2} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2} (31 x+23)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{55} \int \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2} (31 x+23)dx+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{55} \left (\frac {2}{135} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (207 x+1090)}{2 \sqrt {1-2 x}}dx+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (207 x+1090)}{\sqrt {1-2 x}}dx+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (244218 x+162329)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \int \frac {(5 x+3)^{3/2} (244218 x+162329)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (5442127 x+3533916)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (5442127 x+3533916)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {361589456 x+228926353}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5442127}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {361589456 x+228926353}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5442127}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5442127}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5442127}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {3}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {10884254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5442127}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {81406}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {207}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {62}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
(2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/55 + (3*((62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/135 + ((-207*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/35 + ((-81406*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^ (3/2))/5 + ((-5442127*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-36 1589456*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (10884254*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5) /18)/5)/70)/135))/55
3.28.9.3.1 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 = 1.31 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (19136250000 x^{7}+175594749 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-180794728 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+34232625000 x^{6}+2775262500 x^{5}-26590794750 x^{4}-11860640550 x^{3}+4130561505 x^{2}+2535586005 x +72957690\right )}{116943750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(160\) |
| risch | \(\frac {\left (42525000 x^{4}+43470000 x^{3}-17237250 x^{2}-27227430 x -810641\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{7796250 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {228926353 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {90397364 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{214396875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(261\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {302527 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{86625}+\frac {810641 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7796250}+\frac {228926353 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {180794728 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{409303125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {7661 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3465}-\frac {60 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {184 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(284\) |
-1/116943750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(19136250000*x^7+17 5594749*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellipti cF((10+15*x)^(1/2),1/35*70^(1/2))-180794728*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))+3423 2625000*x^6+2775262500*x^5-26590794750*x^4-11860640550*x^3+4130561505*x^2+ 2535586005*x+72957690)/(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 = 69, normalized size of antiderivative = 0.32 \[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=-\frac {1}{7796250} \, {\left (42525000 \, x^{4} + 43470000 \, x^{3} - 17237250 \, x^{2} - 27227430 \, x - 810641\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {6143407141}{10524937500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {90397364}{58471875} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-1/7796250*(42525000*x^4 + 43470000*x^3 - 17237250*x^2 - 27227430*x - 8106 41)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 6143407141/10524937500*sq rt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 90397364/5 8471875*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver se(1159/675, 38998/91125, x + 23/90))
\[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
\[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]