Integrand size = 28, antiderivative size = 218 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=-\frac {2865161 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {231061879 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {3963068 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8859375 \sqrt {33}} \]
-231061879/584718750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2)) *33^(1/2)-3963068/292359375*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155 ^(1/2))*33^(1/2)+62/1485*(1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2)+2/55*(1 -2*x)^(5/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2)+181333/3898125*(2+3*x)^(3/2)*(1-2* x)^(1/2)*(3+5*x)^(1/2)+4258/155925*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)-2865161/19490625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (7167169+9526995 x-23717250 x^2-6142500 x^3+25515000 x^4\right )+231061879 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-238988015 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{584718750} \]
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(7167169 + 9526995*x - 23717 250*x^2 - 6142500*x^3 + 25515000*x^4) + (231061879*I)*Sqrt[33]*EllipticE[I *ArcSinh[Sqrt[9 + 15*x]], -2/33] - (238988015*I)*Sqrt[33]*EllipticF[I*ArcS inh[Sqrt[9 + 15*x]], -2/33])/584718750
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, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^{5/2}}{\sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{55} (1-2 x)^{5/2} (3 x+2)^{5/2} \sqrt {5 x+3}-\frac {2}{55} \int -\frac {5 (1-2 x)^{3/2} (3 x+2)^{3/2} (31 x+23)}{2 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(1-2 x)^{3/2} (3 x+2)^{3/2} (31 x+23)}{\sqrt {5 x+3}}dx+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{135} \int \frac {\sqrt {1-2 x} (3 x+2)^{3/2} (2129 x+2516)}{2 \sqrt {5 x+3}}dx+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \int \frac {\sqrt {1-2 x} (3 x+2)^{3/2} (2129 x+2516)}{\sqrt {5 x+3}}dx+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {2}{105} \int \frac {(172633-181333 x) (3 x+2)^{3/2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \int \frac {(172633-181333 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {181333}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (5730322 x+4243325)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (5730322 x+4243325)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {231061879 x+147355877}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5730322}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {231061879 x+147355877}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5730322}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {43593748}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {231061879}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5730322}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {43593748}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {231061879}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5730322}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {7926136}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {231061879}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5730322}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {181333}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {4258}{105} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {62}{135} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}\) |
(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/55 + ((62*(1 - 2*x)^(3/2 )*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/135 + ((4258*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2 )*Sqrt[3 + 5*x])/105 + ((181333*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x ])/25 + (3*((-5730322*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ((-2 31061879*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (7926136*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5) /15))/50)/105)/135)/11
3.28.86.3.1 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 = 160, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-22963500000 x^{7}+225331557 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-231061879 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-12077100000 x^{6}+30942000000 x^{5}+11093382000 x^{4}-19110351150 x^{3}-7213782660 x^{2}+3219964590 x +1290090420\right )}{584718750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(160\) |
| risch | \(-\frac {\left (25515000 x^{4}-6142500 x^{3}-23717250 x^{2}+9526995 x +7167169\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{19490625 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {147355877 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2143968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {231061879 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2143968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {211711 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{433125}+\frac {7167169 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19490625}+\frac {147355877 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2046515625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {231061879 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2046515625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {21082 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17325}+\frac {72 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{55}-\frac {52 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{165}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
-1/584718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-22963500000*x^7+2 25331557*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellipt icF((10+15*x)^(1/2),1/35*70^(1/2))-231061879*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))-120 77100000*x^6+30942000000*x^5+11093382000*x^4-19110351150*x^3-7213782660*x^ 2+3219964590*x+1290090420)/(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 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\frac {1}{19490625} \, {\left (25515000 \, x^{4} - 6142500 \, x^{3} - 23717250 \, x^{2} + 9526995 \, x + 7167169\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {7947605713}{52624687500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {231061879}{584718750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/19490625*(25515000*x^4 - 6142500*x^3 - 23717250*x^2 + 9526995*x + 716716 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 7947605713/52624687500*sqr t(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 231061879/5 84718750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInve rse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}}{\sqrt {5\,x+3}} \,d x \]