Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {595324 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {18016 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5929 \sqrt {33}} \]
-595324/195657*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-18016/195657*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^ (1/2)+4/77/(3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)+186/539*(1-2*x)^(1/2) /(3+5*x)^(3/2)/(2+3*x)^(1/2)-45040/17787*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5* x)^(3/2)+2976620/195657*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {8473261+10598372 x-32744810 x^2-44649300 x^3}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+2 i \sqrt {33} \left (148831 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-153335 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{195657} \]
(2*((8473261 + 10598372*x - 32744810*x^2 - 44649300*x^3)/(Sqrt[1 - 2*x]*Sq rt[2 + 3*x]*(3 + 5*x)^(3/2)) + (2*I)*Sqrt[33]*(148831*EllipticE[I*ArcSinh[ Sqrt[9 + 15*x]], -2/33] - 153335*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/3 3])))/195657
Time = 0.28 (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 = {115, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {150 x+131}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {150 x+131}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {2}{7} \int \frac {5 (283-279 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \int \frac {283-279 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (-\frac {2}{33} \int \frac {21659-13512 x}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (-\frac {1}{33} \int \frac {21659-13512 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {3 (148831 x+94253)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {297662 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \int \frac {148831 x+94253}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {297662 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {24772}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {148831}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {297662 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {24772}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {148831}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {297662 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{77} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (-\frac {4504}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {148831}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {297662 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {4504 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {186 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}\) |
4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + ((186*Sqrt[1 - 2*x])/ (7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (10*((-4504*Sqrt[1 - 2*x]*Sqrt[2 + 3*x ])/(33*(3 + 5*x)^(3/2)) + ((297662*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (6*((-148831*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5 - (4504*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5))/11)/33))/7)/77
3.30.42.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 = 1.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (1445730 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1488310 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+867438 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-892986 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-44649300 x^{3}-32744810 x^{2}+10598372 x +8473261\right )}{195657 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(219\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {107843}{391314}-\frac {35929 x}{65219}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {10 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {77500}{1331} x^{2}-\frac {38750}{3993} x +\frac {77500}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {754024 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1190648 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
-2/195657*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(1445730*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) -1488310*5^(1/2)*7^(1/2)*EllipticE((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)+867438*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))-8929 86*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((1 0+15*x)^(1/2),1/35*70^(1/2))-44649300*x^3-32744810*x^2+10598372*x+8473261) /(3+5*x)^(3/2)/(6*x^2+x-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 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (44649300 \, x^{3} + 32744810 \, x^{2} - 10598372 \, x - 8473261\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5059657 \, \sqrt {-30} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13394790 \, \sqrt {-30} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{8804565 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
2/8804565*(45*(44649300*x^3 + 32744810*x^2 - 10598372*x - 8473261)*sqrt(5* x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 5059657*sqrt(-30)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*weierstrassPInverse(1159/675, 38998/91125, x + 23/9 0) + 13394790*sqrt(-30)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*weierstra ssZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]