\(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\) [935]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {4}{9} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {107}{27} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {41}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \] Output:

-4/9*(1-2*x)^(1/2)*(3+5*x)^(1/2)-(1-2*x)^(3/2)*(3+5*x)^(1/2)/(6+9*x)-107/1 
35*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-41/27*7^(1/2)*arctan(1/7*( 
1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {107 (2+3 x) \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-5 \left (3 \sqrt {1-2 x} \left (33+73 x+30 x^2\right )+41 \sqrt {7} (2+3 x) \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{135 (2+3 x) \sqrt {3+5 x}} \] Input:

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

Output:

(107*(2 + 3*x)*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 5*( 
3*Sqrt[1 - 2*x]*(33 + 73*x + 30*x^2) + 41*Sqrt[7]*(2 + 3*x)*Sqrt[3 + 5*x]* 
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/(135*(2 + 3*x)*Sqrt[3 + 5* 
x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {108, 27, 171, 27, 175, 64, 104, 217, 223}

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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{3} \int -\frac {\sqrt {1-2 x} (40 x+13)}{2 (3 x+2) \sqrt {5 x+3}}dx-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{6} \int \frac {\sqrt {1-2 x} (40 x+13)}{(3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{6} \left (-\frac {1}{15} \int \frac {5 (214 x+47)}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {1}{3} \int \frac {214 x+47}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {287}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {214}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {287}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {428}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (\frac {574}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {428}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-\frac {428}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {82}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \left (-\frac {214}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {82}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {8}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{3 (3 x+2)}\)

Input:

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

Output:

-1/3*((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x) + ((-8*Sqrt[1 - 2*x]*Sqrt[3 
 + 5*x])/3 + ((-214*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (82*Sq 
rt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/3)/6
 

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 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 108

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

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (11+6 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{9 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (-\frac {107 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{270}+\frac {41 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{54}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(132\)
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (321 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -615 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +214 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-410 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+180 x \sqrt {-10 x^{2}-x +3}+330 \sqrt {-10 x^{2}-x +3}\right )}{270 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\)	\(146\)

Input:

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

Output:

1/9*(-1+2*x)*(3+5*x)^(1/2)*(11+6*x)/(2+3*x)/(-(-1+2*x)*(3+5*x))^(1/2)*((1- 
2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+(-107/270*10^(1/2)*arcsin(20/11*x+1/11)+ 
41/54*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^( 
1/2)))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {41 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 107 \, \sqrt {\frac {2}{5}} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {\frac {2}{5}} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 6 \, {\left (6 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{54 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

-1/54*(41*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)* 
sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 107*sqrt(2/5)*(3*x + 2)*arctan(1/4*sqrt 
(2/5)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 6*(6*x + 
 11)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {107}{270} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {41}{54} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2}{9} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{9 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

-107/270*sqrt(10)*arcsin(20/11*x + 1/11) + 41/54*sqrt(7)*arcsin(37/11*x/ab 
s(3*x + 2) + 20/11/abs(3*x + 2)) - 2/9*sqrt(-10*x^2 - x + 3) - 7/9*sqrt(-1 
0*x^2 - x + 3)/(3*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (83) = 166\).

Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.43 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {41}{540} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {107}{270} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {2}{45} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {154 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \] Input:

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

Output:

41/540*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq 
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 
) - sqrt(22)))) - 107/270*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqr 
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) 
 - sqrt(22)))) - 2/45*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 154/9*sqrt(1 
0)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/( 
sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22) 
)/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 
+ 280)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.83 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {321 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x +214 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )+615 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +410 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-615 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -410 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-90 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -165 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{405 x +270} \] Input:

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

Output:

(321*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x + 214*sqrt(10)*a 
sin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 615*sqrt(7)*atan((sqrt(33) - sq 
rt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 410* 
sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt 
(11))/2))/sqrt(2)) - 615*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( 
 - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 410*sqrt(7)*atan((sqrt(33) 
 + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 9 
0*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 165*sqrt(5*x + 3)*sqrt( - 2*x + 1))/( 
135*(3*x + 2))