Integrand size = 28, antiderivative size = 249 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=-\frac {493825477 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{40540500}-\frac {1865989 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1126125}-\frac {564731 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{2252250}-\frac {2014 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{53625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac {2}{65} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac {16416987253 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18427500 \sqrt {33}}-\frac {493825477 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{18427500 \sqrt {33}} \]
-16416987253/608107500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2 ))*33^(1/2)-493825477/608107500*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33* 1155^(1/2))*33^(1/2)-23/3575*(2+3*x)^(3/2)*(3+5*x)^(7/2)*(1-2*x)^(1/2)+2/6 5*(2+3*x)^(5/2)*(3+5*x)^(7/2)*(1-2*x)^(1/2)-1865989/1126125*(3+5*x)^(3/2)* (1-2*x)^(1/2)*(2+3*x)^(1/2)-564731/2252250*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+ 3*x)^(1/2)-2014/53625*(3+5*x)^(7/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-493825477/ 40540500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-707313559+139824180 x+2626854750 x^2+5075689500 x^3+4299277500 x^4+1403325000 x^5\right )+16416987253 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16910812730 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{608107500} \]
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-707313559 + 139824180*x + 2626854750*x^2 + 5075689500*x^3 + 4299277500*x^4 + 1403325000*x^5) + (1641 6987253*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1691081 2730*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/608107500
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, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}-\frac {2}{65} \int -\frac {(3 x+2)^{3/2} (5 x+3)^{5/2} (23 x+27)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \int \frac {(3 x+2)^{3/2} (5 x+3)^{5/2} (23 x+27)}{\sqrt {1-2 x}}dx+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (-\frac {1}{55} \int -\frac {\sqrt {3 x+2} (5 x+3)^{5/2} (12084 x+7895)}{2 \sqrt {1-2 x}}dx-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \int \frac {\sqrt {3 x+2} (5 x+3)^{5/2} (12084 x+7895)}{\sqrt {1-2 x}}dx-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (-\frac {1}{45} \int -\frac {3 (5 x+3)^{5/2} (564731 x+371788)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \int \frac {(5 x+3)^{5/2} (564731 x+371788)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (-\frac {1}{21} \int -\frac {5 (5 x+3)^{3/2} (22391868 x+14677529)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \int \frac {(5 x+3)^{3/2} (22391868 x+14677529)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (493825477 x+320926341)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (493825477 x+320926341)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {32833974506 x+20786800753}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {493825477}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {32833974506 x+20786800753}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {493825477}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {5432080247}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {32833974506}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {493825477}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {5432080247}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {32833974506}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {493825477}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{110} \left (\frac {1}{15} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {987650954}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {32833974506}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {493825477}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7463956}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {564731}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {4028}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\right )-\frac {23}{55} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}\right )+\frac {2}{65} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}\) |
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/65 + ((-23*Sqrt[1 - 2*x] *(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/55 + ((-4028*Sqrt[1 - 2*x]*Sqrt[2 + 3*x] *(3 + 5*x)^(7/2))/15 + ((-564731*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/ 2))/21 + (5*((-7463956*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + (( -493825477*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-32833974506*S qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (98765095 4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18)/5)) /42)/15)/110)/65
3.27.62.3.1 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 = 1.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-631496250000 x^{8}-2418822000000 x^{7}+15944480574 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16416987253 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3619961887500 x^{6}-2355474127500 x^{5}-49303397250 x^{4}+1002683563200 x^{3}+495121644255 x^{2}-61683747495 x -63658220310\right )}{608107500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(165\) |
| risch | \(-\frac {\left (1403325000 x^{5}+4299277500 x^{4}+5075689500 x^{3}+2626854750 x^{2}+139824180 x -707313559\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{40540500 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {20786800753 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{4459455000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {16416987253 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2229727500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {776801 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{225225}-\frac {707313559 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{40540500}+\frac {20786800753 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{4256752500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16416987253 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2128376250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {89807 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1386}+\frac {450 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}+\frac {15165 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{143}+\frac {53711 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{429}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(306\) |
-1/608107500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-631496250000*x^8- 2418822000000*x^7+15944480574*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))-16416987253*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))-3619961887500*x^6-2355474127500*x^5-49303397250*x^4+10 02683563200*x^3+495121644255*x^2-61683747495*x-63658220310)/(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 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\frac {1}{40540500} \, {\left (1403325000 \, x^{5} + 4299277500 \, x^{4} + 5075689500 \, x^{3} + 2626854750 \, x^{2} + 139824180 \, x - 707313559\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {278907663533}{27364837500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {16416987253}{608107500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/40540500*(1403325000*x^5 + 4299277500*x^4 + 5075689500*x^3 + 2626854750* x^2 + 139824180*x - 707313559)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 278907663533/27364837500*sqrt(-30)*weierstrassPInverse(1159/675, 38998/9 1125, x + 23/90) + 16416987253/608107500*sqrt(-30)*weierstrassZeta(1159/67 5, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]