\(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\) [1537]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {54 \sqrt {1-2 x} \sqrt {3+5 x}}{539 (2+3 x)^{3/2}}+\frac {5256 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}-\frac {1752}{539} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {36}{539} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

4/77*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(3/2)+54/539*(1-2*x)^(1/2)*(3+5*x 
)^(1/2)/(2+3*x)^(3/2)+5256/3773*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)- 
1752/3773*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+ 
36/3773*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 = 6.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (5543-3006 x-15768 x^2\right )}{\sqrt {1-2 x} (2+3 x)^{3/2}}+2 i \sqrt {33} \left (438 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-455 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{3773} \] Input:

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

Output:

(2*((Sqrt[3 + 5*x]*(5543 - 3006*x - 15768*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^( 
3/2)) + (2*I)*Sqrt[33]*(438*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 
455*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/3773
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {115, 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.

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

\(\Big \downarrow \) 115

\(\displaystyle \frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}-\frac {2}{77} \int -\frac {3 (30 x+29)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{77} \int \frac {30 x+29}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {3}{77} \left (\frac {2}{21} \int \frac {3 (116-45 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \int \frac {116-45 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \left (\frac {2}{7} \int \frac {5 (876 x+563)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \left (\frac {5}{7} \int \frac {876 x+563}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {3}{77} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {34}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{3/2}}\)

Input:

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

Output:

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (3*((18*Sqrt[1 - 2* 
x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(3/2)) + (2*((876*Sqrt[1 - 2*x]*Sqrt[3 + 5* 
x])/(7*Sqrt[2 + 3*x]) + (5*((-292*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt 
[1 - 2*x]], 35/33])/5 - (34*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 
 2*x]], 35/33])/5))/7))/7))/77
 

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 115

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]
 

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 [A] (verified)

Time = 0.73 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.34

method	result	size
default	\(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1683 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2628 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1122 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1752 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-78840 x^{3}-62334 x^{2}+18697 x +16629\right )}{3773 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\)	\(215\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {8 \left (-30 x^{2}-38 x -12\right )}{3773 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {5630 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8760 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{147 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {4800}{343} x^{2}-\frac {480}{343} x +\frac {1440}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)

Input:

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

Output:

-2/3773*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1683*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)-2628*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)+1122*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))-1752*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))-788 
40*x^3-62334*x^2+18697*x+16629)/(2+3*x)^(3/2)/(10*x^2+x-3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (15 \, {\left (15768 \, x^{2} + 3006 \, x - 5543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5087 \, \sqrt {-30} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13140 \, \sqrt {-30} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{56595 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \] Input:

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

Output:

2/56595*(15*(15768*x^2 + 3006*x - 5543)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(- 
2*x + 1) - 5087*sqrt(-30)*(18*x^3 + 15*x^2 - 4*x - 4)*weierstrassPInverse( 
1159/675, 38998/91125, x + 23/90) + 13140*sqrt(-30)*(18*x^3 + 15*x^2 - 4*x 
 - 4)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 
 38998/91125, x + 23/90)))/(18*x^3 + 15*x^2 - 4*x - 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{540 x^{6}+864 x^{5}+99 x^{4}-425 x^{3}-154 x^{2}+52 x +24}d x \] Input:

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

Output:

int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(540*x**6 + 864*x**5 + 
99*x**4 - 425*x**3 - 154*x**2 + 52*x + 24),x)