\(\int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [1297]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {92 \sqrt {1-2 x}}{7 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2780 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {556}{3} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {16}{3} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

2/3*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+92/7*(1-2*x)^(1/2)/(2+3*x)^( 
1/2)/(3+5*x)^(1/2)-2780/21*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+556/2 
1*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-16/21*El 
lipticF(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 = 6.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \left (1759+5422 x+4170 x^2\right )}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4 i \left (1529 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1575 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{7 \sqrt {33}} \] Input:

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

Output:

(-2*Sqrt[1 - 2*x]*(1759 + 5422*x + 4170*x^2))/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 
5*x]) - (((4*I)/7)*(1529*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 157 
5*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/Sqrt[33]
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {110, 25, 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.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2}{3} \int -\frac {13-15 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} \int \frac {13-15 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{3} \left (\frac {2}{7} \int \frac {5 (223-138 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \int \frac {223-138 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (-\frac {2}{11} \int \frac {33 (139 x+88)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (-6 \int \frac {139 x+88}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (-6 \left (\frac {23}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {139}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (-6 \left (\frac {23}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {139}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2}{3} \left (\frac {5}{7} \left (-6 \left (-\frac {46 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {139}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}\right )+\frac {138 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{3/2} \sqrt {5 x+3}}\)

Input:

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

Output:

(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (2*((138*Sqrt[1 - 2* 
x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (5*((-278*Sqrt[1 - 2*x]*Sqrt[2 + 3*x 
])/Sqrt[3 + 5*x] - 6*((-139*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 
 2*x]], 35/33])/5 - (46*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]) 
/(5*Sqrt[33]))))/7))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 110

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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 [A] (verified)

Time = 0.40 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (414 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-834 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+276 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-556 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-25020 x^{3}-20022 x^{2}+5712 x +5277\right )}{21 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\)	\(215\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9 \left (\frac {2}{3}+x \right )^{2}}-\frac {346 \left (-30 x^{2}-3 x +9\right )}{21 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1760 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2780 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)

Input:

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

Output:

2/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(414*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)-834*2^(1/2)*E 
llipticE(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)+276*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellip 
ticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-556*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))-25020*x^3- 
20022*x^2+5712*x+5277)/(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 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (4170 \, x^{2} + 5422 \, x + 1759\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 4723 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 12510 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{945 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \] Input:

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

Output:

-2/945*(135*(4170*x^2 + 5422*x + 1759)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2 
*x + 1) - 4723*sqrt(-30)*(45*x^3 + 87*x^2 + 56*x + 12)*weierstrassPInverse 
(1159/675, 38998/91125, x + 23/90) + 12510*sqrt(-30)*(45*x^3 + 87*x^2 + 56 
*x + 12)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/6 
75, 38998/91125, x + 23/90)))/(45*x^3 + 87*x^2 + 56*x + 12)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{675 x^{5}+2160 x^{4}+2763 x^{3}+1766 x^{2}+564 x +72}d x \] Input:

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

Output:

int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(675*x**5 + 2160*x**4 + 
 2763*x**3 + 1766*x**2 + 564*x + 72),x)