\(\int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx\) [1436]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{63 (2+3 x)^{3/2}}-\frac {272 \sqrt {1-2 x} \sqrt {3+5 x}}{441 \sqrt {2+3 x}}+\frac {272}{189} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {206}{189} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

2/63*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-272/441*(1-2*x)^(1/2)*(3+5* 
x)^(1/2)/(2+3*x)^(1/2)+272/1323*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35 
*1155^(1/2))*35^(1/2)-206/1323*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35* 
1155^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (265+408 x)}{(2+3 x)^{3/2}}-136 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1323} \] Input:

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

Output:

(2*((-3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(265 + 408*x))/(2 + 3*x)^(3/2) - (136* 
I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (35*I)*Sqrt[33]* 
EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/1323
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}-\frac {2}{63} \int -\frac {515 x+298}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \int \frac {515 x+298}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{63} \left (\frac {2}{7} \int \frac {5 (59-272 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {272 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{63} \left (\frac {5}{7} \int \frac {59-272 x}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {272 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{63} \left (\frac {5}{7} \left (\frac {1111}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {272}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {272 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{63} \left (\frac {5}{7} \left (\frac {1111}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {272}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {272 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{63} \left (\frac {5}{7} \left (\frac {272}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {202}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {272 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{63 (3 x+2)^{3/2}}\)

Input:

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

Output:

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(63*(2 + 3*x)^(3/2)) + ((-272*Sqrt[1 - 2*x 
]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (5*((272*Sqrt[11/3]*EllipticE[ArcSin[ 
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (202*Sqrt[11/3]*EllipticF[ArcSin[Sqr 
t[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/7)/63
 

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 169

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]
 

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 = 0.62 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {\left (9999 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+816 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+6666 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+544 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+24480 x^{3}+18348 x^{2}-5754 x -4770\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1323 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\)	\(215\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {295 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9261 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9261 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567 \left (\frac {2}{3}+x \right )^{2}}-\frac {272 \left (-30 x^{2}-3 x +9\right )}{1323 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(225\)

Input:

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

Output:

-1/1323*(9999*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)+816*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)+6666*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))+544*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ell 
ipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+24480*x^3+18348*x^2-5754*x-4770)* 
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx=-\frac {270 \, {\left (408 \, x + 265\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 5783 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 12240 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{59535 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

-1/59535*(270*(408*x + 265)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 5 
783*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInverse(1159/675, 38998/91125 
, x + 23/90) + 12240*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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx=\frac {-30 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-7875 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}-10500 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x -3500 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )+2538 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}+3384 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x +1128 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )}{153 x^{2}+204 x +68} \] Input:

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

Output:

( - 30*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 7875*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**2 - 10500*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sq 
rt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 2 
4),x)*x - 3500*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(27 
0*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x) + 2538*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 + 3384*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 
 + 1128*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))/(17*(9*x**2 + 12*x + 4))