Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {2632}{243} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {614}{27} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}-\frac {9587 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1215}+\frac {2632 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1215} \]
-2/9*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(3/2)-9587/3645*EllipticE(1/7*21^ (1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2632/3645*EllipticF(1/7*21^( 1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+362/27*(3+5*x)^(5/2)*(1-2*x)^ (1/2)/(2+3*x)^(1/2)-614/27*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+2632/ 243*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.93 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (1187+2463 x+468 x^2-810 x^3\right )}{(2+3 x)^{3/2}}+9587 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-6955 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3645} \]
((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1187 + 2463*x + 468*x^2 - 810*x^3))/(2 + 3*x)^(3/2) + (9587*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33 ] - (6955*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/3645
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {108, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{9} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \int \frac {(1307-4605 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1}{15} \int \frac {15 (447-2632 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1}{2} \int \frac {(447-2632 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{9} \int -\frac {9587 x+2857}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2632}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{2} \left (-\frac {1}{9} \int \frac {9587 x+2857}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2632}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{9} \left (\frac {14476}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9587}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2632}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{9} \left (\frac {14476}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9587}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2632}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{9} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{9} \left (\frac {9587}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2632}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {2632}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+307 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) + ((362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) - (2*(307*Sqrt[1 - 2*x]*Sqrt[2 + 3 *x]*(3 + 5*x)^(3/2) + ((-2632*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((9587*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2632*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9 )/2))/3)/9
3.28.23.3.1 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[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 = 1.25 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {\left (137709 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-201327 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+91806 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-134218 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+1701000 x^{5}-812700 x^{4}-5780880 x^{3}-2715090 x^{2}+1302420 x +747810\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{25515 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) | \(229\) |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {20 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}+\frac {344 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243}+\frac {5714 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {19174 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {5060}{243} x^{2}-\frac {506}{243} x +\frac {506}{81}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{2}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(269\) |
-1/25515*(137709*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))* x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)-201327*5^(1/2)*7^(1/2)*Ellipt icE((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^ (1/2)+91806*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ell ipticF((10+15*x)^(1/2),1/35*70^(1/2))-134218*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))+170 1000*x^5-812700*x^4-5780880*x^3-2715090*x^2+1302420*x+747810)*(3+5*x)^(1/2 )*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=-\frac {2700 \, {\left (810 \, x^{3} - 468 \, x^{2} - 2463 \, x - 1187\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 36629 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 862830 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{328050 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
-1/328050*(2700*(810*x^3 - 468*x^2 - 2463*x - 1187)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 36629*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInve rse(1159/675, 38998/91125, x + 23/90) - 862830*sqrt(-30)*(9*x^2 + 12*x + 4 )*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 389 98/91125, x + 23/90)))/(9*x^2 + 12*x + 4)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \]