Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {95264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{250047} \]
-2/27*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(9/2)+230/567*(1-2*x)^(3/2)*(3+5 *x)^(3/2)/(2+3*x)^(7/2)-3545996/750141*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2 ),1/33*1155^(1/2))*33^(1/2)-95264/750141*EllipticF(1/7*21^(1/2)*(1-2*x)^(1 /2),1/33*1155^(1/2))*33^(1/2)+1532/567*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x) ^(5/2)-104036/35721*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+3545996/2500 47*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 9.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (29785139+176436240 x+391601529 x^2+386630766 x^3+143612838 x^4\right )}{2 (2+3 x)^{9/2}}+i \sqrt {33} \left (886499 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-910315 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{750141} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(29785139 + 176436240*x + 391601529*x^2 + 386630766*x^3 + 143612838*x^4))/(2*(2 + 3*x)^(9/2)) + I*Sqrt[33]*(88649 9*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 910315*EllipticF[I*ArcSinh [Sqrt[9 + 15*x]], -2/33])))/750141
Time = 0.29 (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, 167, 27, 167, 27, 167, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{27} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)^{9/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{(3 x+2)^{9/2}}dx-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \int \frac {3 \sqrt {1-2 x} (127-x) \sqrt {5 x+3}}{(3 x+2)^{7/2}}dx-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \int \frac {\sqrt {1-2 x} (127-x) \sqrt {5 x+3}}{(3 x+2)^{7/2}}dx-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}-\frac {2}{15} \int -\frac {(6123-3820 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{5/2}}dx\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \int \frac {(6123-3820 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{5/2}}dx+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{63} \int \frac {201493-141010 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \int \frac {201493-141010 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \left (\frac {2}{7} \int \frac {5 (886499 x+558097)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1772998 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \left (\frac {10}{7} \int \frac {886499 x+558097}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1772998 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \left (\frac {10}{7} \left (\frac {130988}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {886499}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {1772998 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \left (\frac {10}{7} \left (\frac {130988}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {886499}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1772998 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {1}{63} \left (\frac {10}{7} \left (-\frac {23816}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {886499}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1772998 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {52018 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\right )+\frac {766 \sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\right )-\frac {46 (1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) - (5*((-46*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(21*(2 + 3*x)^(7/2)) - (2*((766*Sqrt[1 - 2*x]* (3 + 5*x)^(3/2))/(15*(2 + 3*x)^(5/2)) + ((-52018*Sqrt[1 - 2*x]*Sqrt[3 + 5* x])/(63*(2 + 3*x)^(3/2)) + ((1772998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[ 2 + 3*x]) + (10*((-886499*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2 *x]], 35/33])/5 - (23816*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5))/7)/63)/15))/7))/27
3.28.72.3.1 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 = 1.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.36
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {776 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59049 \left (\frac {2}{3}+x \right )^{4}}+\frac {5930 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137781 \left (\frac {2}{3}+x \right )^{3}}+\frac {38764 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {35459960}{250047} x^{2}-\frac {3545996}{250047} x +\frac {3545996}{83349}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {4464776 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5250987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {7091992 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{5250987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{5}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(302\) |
| default | \(-\frac {2 \left (139044114 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-143612838 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+370784304 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-382967568 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+370784304 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-382967568 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+164793024 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-170207808 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+27465504 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-28367968 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4308385140 x^{6}-12029761494 x^{5}-11615422626 x^{4}-2988214893 x^{3}+2101550871 x^{2}+1498570743 x +268066251\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{750141 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(504\) |
-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5* x)*(2+3*x))^(1/2)*(-776/59049*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+5930/ 137781*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+38764/321489*(-30*x^3-23*x^2 +7*x+6)^(1/2)/(2/3+x)^2+3545996/750141*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3 *x+9))^(1/2)+4464776/5250987*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/ 2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+7 091992/5250987*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23 *x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*Ellip ticF((10+15*x)^(1/2),1/35*70^(1/2)))+98/177147*(-30*x^3-23*x^2+7*x+6)^(1/2 )/(2/3+x)^5)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (143612838 \, x^{4} + 386630766 \, x^{3} + 391601529 \, x^{2} + 176436240 \, x + 29785139\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 29839253 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 79784910 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{33756345 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
2/33756345*(135*(143612838*x^4 + 386630766*x^3 + 391601529*x^2 + 176436240 *x + 29785139)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 29839253*sqrt( -30)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassPInv erse(1159/675, 38998/91125, x + 23/90) + 79784910*sqrt(-30)*(243*x^5 + 810 *x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassZeta(1159/675, 38998/91 125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(243*x^5 + 81 0*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]