Integrand size = 28, antiderivative size = 156 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {4282 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7875}+\frac {118}{525} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{21} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {86741 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{3375 \sqrt {35}}+\frac {16088 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{3375 \sqrt {35}} \] Output:
4282/7875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+118/525*(1-2*x)^(3/2)* (2+3*x)^(1/2)*(3+5*x)^(1/2)+2/21*(1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2) -86741/118125*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1 /2)+16088/118125*EllipticF(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 = 3.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (3401-3270 x+1500 x^2\right )+86741 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-74935 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{118125} \] Input:
Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/Sqrt[2 + 3*x],x]
Output:
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3401 - 3270*x + 1500*x^2) + (86741*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (74935*I )*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/118125
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {112, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{\sqrt {3 x+2}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{21} (1-2 x)^{5/2} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2}{21} \int -\frac {(1-2 x)^{3/2} (177 x+104)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \int \frac {(1-2 x)^{3/2} (177 x+104)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{21} \left (\frac {2}{75} \int \frac {9 \sqrt {1-2 x} (2141 x+1201)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \int \frac {\sqrt {1-2 x} (2141 x+1201)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \left (\frac {2}{45} \int \frac {86741 x+39058}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4282}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \left (\frac {1}{45} \int \frac {86741 x+39058}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4282}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \left (\frac {1}{45} \left (\frac {86741}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {64933}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {4282}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \left (\frac {1}{45} \left (-\frac {64933}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {86741}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {4282}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{21} \left (\frac {3}{25} \left (\frac {1}{45} \left (\frac {11806}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {86741}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {4282}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {118}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {2}{21} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{5/2}\) |
Input:
Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/Sqrt[2 + 3*x],x]
Output:
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/21 + ((118*(1 - 2*x)^(3/2) *Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/25 + (3*((4282*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*S qrt[3 + 5*x])/45 + ((-86741*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (11806*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/45))/25)/21
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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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 = 0.38 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (194799 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+86741 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1350000 x^{5}-1908000 x^{4}+489600 x^{3}+2763390 x^{2}-125610 x -612180\right )}{3543750 x^{3}+2716875 x^{2}-826875 x -708750}\) | \(148\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {436 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{525}+\frac {6802 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7875}+\frac {39058 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{165375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {86741 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{165375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
| risch | \(-\frac {2 \left (1500 x^{2}-3270 x +3401\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{7875 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {39058 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{433125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {86741 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{433125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(252\) |
Input:
int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/118125*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(194799*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))+86741*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))+1350000*x^5-1908000*x^4+489600*x^3+276339 0*x^2-125610*x-612180)/(30*x^3+23*x^2-7*x-6)
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.38 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {2}{7875} \, {\left (1500 \, x^{2} - 3270 \, x + 3401\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {1520177}{10631250} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {86741}{118125} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="fricas")
Output:
2/7875*(1500*x^2 - 3270*x + 3401)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1520177/10631250*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 86741/118125*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125 , weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\sqrt {3 x + 2}}\, dx \] Input:
integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**(1/2),x)
Output:
Integral((1 - 2*x)**(5/2)*sqrt(5*x + 3)/sqrt(3*x + 2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/sqrt(3*x + 2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/sqrt(3*x + 2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{\sqrt {3\,x+2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(1/2),x)
Output:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(1/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {8 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{21}-\frac {436 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{525}+\frac {1549 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{4025}+\frac {86741 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{4025}-\frac {53419 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{8050} \] Input:
int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x)
Output:
(9200*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 20056*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 9294*sqrt(3*x + 2)*sqrt(5*x + 3)*sq rt( - 2*x + 1) + 520446*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)* x**2)/(30*x**3 + 23*x**2 - 7*x - 6),x) - 160257*int((sqrt(3*x + 2)*sqrt(5* x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/24150