\(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx\) [1564]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {194 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 \sqrt {2+3 x}}+\frac {458 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{1617}-\frac {194 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{1617} \] Output:

2/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(1/2)+194/1617*(3+5*x)^(1/2)/(1-2 
*x)^(1/2)/(2+3*x)^(1/2)-458/3773*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2) 
+458/11319*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2) 
-194/11319*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 = 6.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (531+1390 x-2748 x^2\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}}-i \sqrt {33} \left (229 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-140 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{11319} \] Input:

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

Output:

(2*((Sqrt[3 + 5*x]*(531 + 1390*x - 2748*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3* 
x]) - I*Sqrt[33]*(229*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 140*El 
lipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/11319
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {110, 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.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}-\frac {2}{21} \int -\frac {45 x+26}{2 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \int \frac {45 x+26}{(1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{21} \left (\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}-\frac {2}{77} \int -\frac {3 (485 x+247)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \int \frac {485 x+247}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \left (\frac {2}{7} \int -\frac {5 (458 x+79)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \left (-\frac {5}{7} \int \frac {458 x+79}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \left (-\frac {5}{7} \left (\frac {458}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {979}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \left (-\frac {5}{7} \left (-\frac {979}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {458}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{21} \left (\frac {3}{77} \left (-\frac {5}{7} \left (\frac {178}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {458}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {194 \sqrt {5 x+3}}{77 \sqrt {1-2 x} \sqrt {3 x+2}}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} \sqrt {3 x+2}}\)

Input:

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

Output:

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) + ((194*Sqrt[3 + 5*x] 
)/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (3*((-458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] 
)/(7*Sqrt[2 + 3*x]) - (5*((-458*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt 
[1 - 2*x]], 35/33])/5 + (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 
- 2*x]], 35/33])/5))/7))/77)/21
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(116)=232\).

Time = 0.73 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{147 \left (x -\frac {1}{2}\right )^{2}}-\frac {260 \left (-30 x^{2}-38 x -12\right )}{11319 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {395 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{79233 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2290 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{79233 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6 \left (-30 x^{2}-3 x +9\right )}{343 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)
default	\(-\frac {\left (123354 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x +19236 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -61677 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )-9618 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+577080 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}+54348 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}-286650 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x +2007180 x^{3}+1538838 x^{2}-468342 x -401436\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{237699 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(359\)

Input:

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

Output:

(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 
)*(1/147*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2-260/11319*(-30*x^2-38*x-12 
)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)-395/79233*(28+42*x)^(1/2)*(-15*x-9)^(1 
/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF(1/7*(28+42*x)^( 
1/2),1/2*70^(1/2))-2290/79233*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1 
/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-1/15*EllipticE(1/7*(28+42*x)^(1/2),1/2* 
70^(1/2))-3/5*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2)))-6/343*(-30*x^2- 
3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (2748 \, x^{2} - 1390 \, x - 531\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 856 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 10305 \, \sqrt {-30} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{509355 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \] Input:

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

Output:

-2/509355*(45*(2748*x^2 - 1390*x - 531)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(- 
2*x + 1) + 856*sqrt(-30)*(12*x^3 - 4*x^2 - 5*x + 2)*weierstrassPInverse(11 
59/675, 38998/91125, x + 23/90) + 10305*sqrt(-30)*(12*x^3 - 4*x^2 - 5*x + 
2)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38 
998/91125, x + 23/90)))/(12*x^3 - 4*x^2 - 5*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{72 x^{5}-12 x^{4}-58 x^{3}+15 x^{2}+12 x -4}d x \right ) \] Input:

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

Output:

 - int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(72*x**5 - 12*x**4 - 
 58*x**3 + 15*x**2 + 12*x - 4),x)