Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {310399 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{382725} \]
-25111/1148175*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-310399/1148175*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)-2/3*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(1/2)-40/81*(1-2*x)^(3/2)* (3+5*x)^(5/2)*(2+3*x)^(1/2)+64628/8505*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x) ^(1/2)-2108/1701*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-310399/76545*(1 -2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\frac {\frac {15 \sqrt {1-2 x} \sqrt {3+5 x} \left (21964+245751 x-259650 x^2-386100 x^3+567000 x^4\right )}{\sqrt {2+3 x}}+25111 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-335510 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1148175} \]
((15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(21964 + 245751*x - 259650*x^2 - 386100*x ^3 + 567000*x^4))/Sqrt[2 + 3*x] + (25111*I)*Sqrt[33]*EllipticE[I*ArcSinh[S qrt[9 + 15*x]], -2/33] - (335510*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/1148175
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {108, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{3} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 \sqrt {3 x+2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{3} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{\sqrt {3 x+2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{3} \left (\frac {2}{135} \int -\frac {5 (65-1054 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 \sqrt {3 x+2}}dx+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{3} \left (\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1}{27} \int \frac {(65-1054 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{\sqrt {3 x+2}}dx\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2}{105} \int \frac {(56363-193884 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1}{105} \int \frac {(56363-193884 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{15} \int \frac {3 (27033-310399 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \int \frac {(27033-310399 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \left (\frac {310399}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{9} \int \frac {50222 x+713011}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \left (\frac {310399}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{18} \int \frac {50222 x+713011}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {3414389}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {50222}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {310399}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {50222}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {3414389}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {310399}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{3} \left (\frac {1}{27} \left (\frac {1}{105} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {620798}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {50222}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {310399}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {64628}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {2108}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8}{27} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) - (5*((8*(1 - 2*x)^ (3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/27 + ((2108*Sqrt[1 - 2*x]*Sqrt[2 + 3* x]*(3 + 5*x)^(5/2))/105 + ((-64628*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^( 3/2))/5 + ((310399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((50222* Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (620798* Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18)/5)/10 5)/27))/3
3.28.78.3.1 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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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 = 1.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (316338 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-25111 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-85050000 x^{6}+49410000 x^{5}+70254000 x^{4}-50342400 x^{3}-18665115 x^{2}+10729335 x +988380\right )}{1148175 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(155\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {1850 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701}+\frac {26417 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{76545}+\frac {713011 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{8037225 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {50222 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{8037225 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5660 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701}+\frac {200 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}-\frac {98 \left (-30 x^{2}-3 x +9\right )}{729 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
-1/1148175*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(316338*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))-25111*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))-85050000*x^6+49410000*x^5+702 54000*x^4-50342400*x^3-18665115*x^2+10729335*x+988380)/(30*x^3+23*x^2-7*x- 6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.40 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\frac {675 \, {\left (567000 \, x^{4} - 386100 \, x^{3} - 259650 \, x^{2} + 245751 \, x + 21964\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 15753971 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1129995 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{51667875 \, {\left (3 \, x + 2\right )}} \]
1/51667875*(675*(567000*x^4 - 386100*x^3 - 259650*x^2 + 245751*x + 21964)* sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 15753971*sqrt(-30)*(3*x + 2)* weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1129995*sqrt(-30)* (3*x + 2)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/ 675, 38998/91125, x + 23/90)))/(3*x + 2)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]