Integrand size = 28, antiderivative size = 187 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {416 \sqrt {1-2 x}}{21 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {2780 \sqrt {1-2 x} \sqrt {2+3 x}}{21 (3+5 x)^{3/2}}+\frac {184840 \sqrt {1-2 x} \sqrt {2+3 x}}{231 \sqrt {3+5 x}}-\frac {36968 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7 \sqrt {33}}-\frac {1112 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{7 \sqrt {33}} \]
-36968/231*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)- 1112/231*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/ 3*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+416/21*(1-2*x)^(1/2)/(3+5*x)^( 3/2)/(2+3*x)^(1/2)-2780/21*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+18484 0/231*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2}{231} \left (\frac {\sqrt {1-2 x} \left (1052533+4998904 x+7902930 x^2+4158900 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (4621 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4760 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
(2*((Sqrt[1 - 2*x]*(1052533 + 4998904*x + 7902930*x^2 + 4158900*x^3))/((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(4621*EllipticE[I*ArcSinh[S qrt[9 + 15*x]], -2/33] - 4760*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]) ))/231
Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {110, 25, 169, 27, 169, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac {2}{3} \int -\frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} \int \frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{3} \left (\frac {2}{7} \int \frac {15 (181-208 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \int \frac {181-208 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{33} \int \frac {11 (674-417 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \int \frac {674-417 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \left (-\frac {2}{11} \int \frac {3 (9242 x+5851)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \left (-\frac {3}{11} \int \frac {9242 x+5851}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9242}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{3} \left (\frac {15}{7} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (-\frac {278}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}\right )+\frac {208 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (2*((208*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (15*((-278*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3 + 5*x)^(3/2)) - (2*((-9242*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11* Sqrt[3 + 5*x]) - (3*((-9242*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (278*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5))/11))/3))/7))/3
3.27.97.3.1 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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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 + 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 = 1.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {38}{675}-\frac {4 x}{45}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {175666}{3465}-\frac {18484 x}{231}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {46808 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1617 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {73936 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1617 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (269280 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-277260 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+341088 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-351196 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+107712 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-110904 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-8317800 x^{4}-11646960 x^{3}-2094878 x^{2}+2893838 x +1052533\right )}{231 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(311\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*((-38/675-4/45*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x^2+19/15*x+2/5)^2-2*(15 -30*x)*(-175666/3465-18484/231*x)/((x^2+19/15*x+2/5)*(15-30*x))^(1/2)+4680 8/1617*(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))+73936/1617*(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*Ell ipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*7 0^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (4158900 \, x^{3} + 7902930 \, x^{2} + 4998904 \, x + 1052533\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 314024 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 831780 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{10395 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
2/10395*(45*(4158900*x^3 + 7902930*x^2 + 4998904*x + 1052533)*sqrt(5*x + 3 )*sqrt(3*x + 2)*sqrt(-2*x + 1) - 314024*sqrt(-30)*(225*x^4 + 570*x^3 + 541 *x^2 + 228*x + 36)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 831780*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*weierstrassZe ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 2 3/90)))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]