Integrand size = 28, antiderivative size = 191 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)^{7/2}}-\frac {536 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{5/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 \sqrt {2+3 x}}-\frac {184636 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{252105}-\frac {9124 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{252105} \]
-184636/756315*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-9124/756315*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)+2/147*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-536/5145*(1-2*x)^(1/2 )*(3+5*x)^(1/2)/(2+3*x)^(5/2)+974/36015*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x )^(3/2)+184636/252105*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.91 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (727631+3324960 x+5015853 x^2+2492586 x^3\right )}{2 (2+3 x)^{7/2}}+i \sqrt {33} \left (46159 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-48440 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{756315} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(727631 + 3324960*x + 5015853*x^2 + 249 2586*x^3))/(2*(2 + 3*x)^(7/2)) + I*Sqrt[33]*(46159*EllipticE[I*ArcSinh[Sqr t[9 + 15*x]], -2/33] - 48440*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])) )/756315
Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {109, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}-\frac {2}{147} \int -\frac {1175 x+694}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{147} \int \frac {1175 x+694}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{147} \left (\frac {2}{35} \int \frac {3 (2680 x+1949)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \int \frac {2680 x+1949}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {13763-2435 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (92318 x+60409)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {92318 x+60409}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {25091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {92318}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {25091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {92318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {4562}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {92318}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {92318 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {536 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\) |
(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) + ((-536*Sqrt[1 - 2* x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (3*((974*Sqrt[1 - 2*x]*Sqrt[3 + 5 *x])/(21*(2 + 3*x)^(3/2)) + (2*((92318*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqr t[2 + 3*x]) + (5*((-92318*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2 *x]], 35/33])/5 - (4562*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5))/7))/21))/35)/147
3.29.29.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 1.32 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.40
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11907 \left (\frac {2}{3}+x \right )^{4}}-\frac {536 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{138915 \left (\frac {2}{3}+x \right )^{3}}+\frac {974 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{324135 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {369272}{50421} x^{2}-\frac {184636}{252105} x +\frac {184636}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {241636 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5294205 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {369272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{5294205 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| default | \(-\frac {2 \left (2466288 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2492586 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4932576 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4985172 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3288384 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3323448 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+730752 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-738544 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-74777580 x^{5}-157953348 x^{4}-92363085 x^{3}+13338867 x^{2}+27741747 x +6548679\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{756315 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(409\) |
(-(-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 )*(2/11907*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-536/138915*(-30*x^3-23*x ^2+7*x+6)^(1/2)/(2/3+x)^3+974/324135*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^ 2+184636/756315*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+241636/529 4205*(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))+369272/5294205*(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*E llipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35 *70^(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 = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\frac {2 \, {\left (135 \, {\left (2492586 \, x^{3} + 5015853 \, x^{2} + 3324960 \, x + 727631\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1656748 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4154310 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{34034175 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
2/34034175*(135*(2492586*x^3 + 5015853*x^2 + 3324960*x + 727631)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1656748*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 4154310*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassZ eta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{9/2}} \,d x \]