Integrand size = 28, antiderivative size = 155 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {133}{10} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+3 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {4621}{30} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {133}{30} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
133/10*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+3*(1-2*x)^(1/2)*(2+3*x)^( 3/2)*(3+5*x)^(1/2)+(2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(1/2)+4621/150*Elli pticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-133/150*Ellipt icF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 6.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {-15 \sqrt {2+3 x} \sqrt {3+5 x} \left (-253+106 x+30 x^2\right )-4621 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4760 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{150 \sqrt {1-2 x}} \] Input:
Integrate[((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(1 - 2*x)^(3/2),x]
Output:
(-15*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-253 + 106*x + 30*x^2) - (4621*I)*Sqrt[3 3 - 66*x]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (4760*I)*Sqrt[33 - 66*x]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(150*Sqrt[1 - 2*x])
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\int \frac {3 \sqrt {3 x+2} \sqrt {5 x+3} (30 x+19)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \int \frac {\sqrt {3 x+2} \sqrt {5 x+3} (30 x+19)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (-\frac {1}{25} \int -\frac {5 \sqrt {5 x+3} (417 x+271)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (417 x+271)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {3 (9242 x+5851)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {139}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \int \frac {9242 x+5851}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {139}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9242}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {139}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {139}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (-\frac {278}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {139}{3} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {6}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )\) |
Input:
Int[((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/(1 - 2*x)^(3/2),x]
Output:
((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] - (3*((-6*Sqrt[1 - 2*x]*Sq rt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-139*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*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)/6)/5))/2
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[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 = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4587 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-9242 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13500 x^{4}+64800 x^{3}-48030 x^{2}-125130 x -45540\right )}{9000 x^{3}+6900 x^{2}-2100 x -1800}\) | \(143\) |
| elliptic | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {3 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2}+\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{20}-\frac {5851 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{420 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4621 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {77 \left (-30 x^{2}-38 x -12\right )}{8 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\) | \(252\) |
Input:
int((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/300*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4587*2^(1/2)*(2+3*x)^(1/2 )*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2)) -9242*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28 +42*x)^(1/2),1/2*70^(1/2))+13500*x^4+64800*x^3-48030*x^2-125130*x-45540)/( 30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {675 \, {\left (30 \, x^{2} + 106 \, x - 253\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 78506 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 207945 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{6750 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
1/6750*(675*(30*x^2 + 106*x - 253)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 78506*sqrt(-30)*(2*x - 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 207945*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/675, 38998/9 1125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2*x - 1)
\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((2+3*x)**(3/2)*(3+5*x)**(3/2)/(1-2*x)**(3/2),x)
Output:
Integral((3*x + 2)**(3/2)*(5*x + 3)**(3/2)/(1 - 2*x)**(3/2), x)
\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(3/2)/(-2*x + 1)^(3/2), x)
\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(3/2)/(-2*x + 1)^(3/2), x)
Timed out. \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \] Input:
int(((3*x + 2)^(3/2)*(5*x + 3)^(3/2))/(1 - 2*x)^(3/2),x)
Output:
int(((3*x + 2)^(3/2)*(5*x + 3)^(3/2))/(1 - 2*x)^(3/2), x)
\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {60 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+212 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -670 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+41888 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right ) x -20944 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right )-16786 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right ) x +8393 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right )}{40 x -20} \] Input:
int((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x)
Output:
(60*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 212*sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 670*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 41888*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/ (60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x)*x - 20944*int((sqrt(3*x + 2)*sq rt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6) ,x) - 16786*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x)*x + 8393*int((sqrt(3*x + 2)*sqrt(5*x + 3)* sqrt( - 2*x + 1))/(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x))/(20*(2*x - 1 ))