\(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\) [1400]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {812 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {3896}{81} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {68}{81} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

14/9*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+812/27*(1-2*x)^(1/2)*(3+5*x 
)^(1/2)/(2+3*x)^(1/2)-3896/405*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35* 
1155^(1/2))*35^(1/2)+68/405*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*115 
5^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2}{405} \left (\frac {735 \sqrt {1-2 x} \sqrt {3+5 x} (17+24 x)}{(2+3 x)^{3/2}}+1948 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-2030 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*Sqrt[3 + 5*x]),x]
 

Output:

(2*((735*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(17 + 24*x))/(2 + 3*x)^(3/2) + (1948* 
I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (2030*I)*Sqrt[33 
]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/405
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {109, 167, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{5/2} \sqrt {5 x+3}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {2}{9} \int \frac {\sqrt {1-2 x} (41 x+95)}{(3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2}{9} \left (\frac {406 \sqrt {1-2 x} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}-\frac {2}{3} \int -\frac {1948 x+1259}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {1}{3} \int \frac {1948 x+1259}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {406 \sqrt {1-2 x} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2}{9} \left (\frac {1}{3} \left (\frac {451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1948}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {406 \sqrt {1-2 x} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2}{9} \left (\frac {1}{3} \left (\frac {451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1948}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {406 \sqrt {1-2 x} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2}{9} \left (\frac {1}{3} \left (-\frac {82}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1948}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {406 \sqrt {1-2 x} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}\)

Input:

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*Sqrt[3 + 5*x]),x]
 

Output:

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (2*((406*Sqrt[1 - 
 2*x]*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) + ((-1948*Sqrt[11/3]*EllipticE[ArcS 
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (82*Sqrt[11/3]*EllipticF[ArcSin[S 
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/3))/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 109

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])
 

rule 123

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])
 

rule 129

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]))
 

rule 167

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]
 

rule 176

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(93)=186\).

Time = 0.38 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {2 \left (4059 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5844 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2706 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3896 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-176400 x^{3}-142590 x^{2}+40425 x +37485\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{405 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\)	\(215\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7840}{27} x^{2}-\frac {784}{27} x +\frac {784}{9}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2518 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {3896 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(225\)

Input:

int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/405*(4059*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x) 
^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-5844*2^(1/2)*EllipticE(1/7*(28+42*x)^( 
1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+2706*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))-3896*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*El 
lipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-176400*x^3-142590*x^2+40425*x+37 
485)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (33075 \, {\left (24 \, x + 17\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 34253 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 87660 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{18225 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")
 

Output:

2/18225*(33075*(24*x + 17)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 34 
253*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInverse(1159/675, 38998/91125 
, x + 23/90) + 87660*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassZeta(1159/675 
, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(9* 
x^2 + 12*x + 4)
 

Sympy [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \] Input:

integrate((1-2*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(1/2),x)
 

Output:

Integral((1 - 2*x)**(5/2)/((3*x + 2)**(5/2)*sqrt(5*x + 3)), x)
 

Maxima [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="maxima")
 

Output:

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(5/2)), x)
 

Giac [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="giac")
 

Output:

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \] Input:

int((1 - 2*x)^(5/2)/((3*x + 2)^(5/2)*(5*x + 3)^(1/2)),x)
 

Output:

int((1 - 2*x)^(5/2)/((3*x + 2)^(5/2)*(5*x + 3)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {136 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +30 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-72450 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}-96600 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x -32200 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )+1134 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}+1512 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x +504 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )}{2295 x^{2}+3060 x +1020} \] Input:

int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2),x)
 

Output:

(2*(68*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 15*sqrt(3*x + 2)*s 
qrt(5*x + 3)*sqrt( - 2*x + 1) - 36225*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqr 
t( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24 
),x)*x**2 - 48300*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/ 
(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x - 16100*int(( 
sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 
333*x**3 - 46*x**2 - 100*x - 24),x) + 567*int((sqrt(3*x + 2)*sqrt(5*x + 3) 
*sqrt( - 2*x + 1))/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24) 
,x)*x**2 + 756*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(270*x** 
5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x + 252*int((sqrt(3*x + 
 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(270*x**5 + 567*x**4 + 333*x**3 - 46*x 
**2 - 100*x - 24),x)))/(255*(9*x**2 + 12*x + 4))