\(\int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx\) [1029]

Optimal result

Integrand size = 26, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {174591 (1-2 x)^{7/2} \sqrt {3+5 x}}{160000}+\frac {2691 (1-2 x)^{9/2} \sqrt {3+5 x}}{4000}-\frac {9}{80} (1-2 x)^{11/2} \sqrt {3+5 x}+\frac {368012183 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000000 \sqrt {10}} \] Output:

33455653/64000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+3041423/19200000*(1-2*x)^(3/ 
2)*(3+5*x)^(1/2)+276493/4800000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-174591/160000* 
(1-2*x)^(7/2)*(3+5*x)^(1/2)+2691/4000*(1-2*x)^(9/2)*(3+5*x)^(1/2)-9/80*(1- 
2*x)^(11/2)*(3+5*x)^(1/2)[phone]/640000000*arcsin(1/11*22^(1/2)*(3+5*x) 
^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=\frac {10 \sqrt {1-2 x} \left (119699127+1203430125 x+971700740 x^2-3357104800 x^3-2630448000 x^4+3767040000 x^5+3456000000 x^6\right )-1104036549 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1920000000 \sqrt {3+5 x}} \] Input:

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

Output:

(10*Sqrt[1 - 2*x]*(119699127 + 1203430125*x + 971700740*x^2 - 3357104800*x 
^3 - 2630448000*x^4 + 3767040000*x^5 + 3456000000*x^6) - 1104036549*Sqrt[3 
0 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(1920000000*Sqrt[3 + 5*x] 
)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {111, 27, 164, 60, 60, 60, 64, 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.

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

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{60} \int -\frac {3 (1-2 x)^{5/2} (3 x+2) (197 x+122)}{2 \sqrt {5 x+3}}dx-\frac {1}{20} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \int \frac {(1-2 x)^{5/2} (3 x+2) (197 x+122)}{\sqrt {5 x+3}}dx-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{40} \left (\frac {276493 \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {276493 \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {276493 \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {276493 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{40} \left (\frac {276493 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{40} \left (\frac {276493 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (47280 x+52951)}{4000}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\)

Input:

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

Output:

-1/20*((1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (-1/4000*((1 - 2*x)^(7 
/2)*Sqrt[3 + 5*x]*(52951 + 47280*x)) + (276493*(((1 - 2*x)^(5/2)*Sqrt[3 + 
5*x])/15 + (11*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*S 
qrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20) 
)/6))/8000)/40
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\left (691200000 x^{5}[phone] x^{4}-729302400 x^{3}-233839520 x^{2}[phone] x [phone]\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{192000000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {368012183 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13824000000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-14586048000 x^{3} \sqrt {-10 x^{2}-x +3}-4676790400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{3840000000 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

Input:

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

Output:

-1/192000000*(691200000*x^5+338688000*x^4-729302400*x^3-233839520*x^2+3346 
43860*x[phone])*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x 
)*(3+5*x))^(1/2)/(1-2*x)^(1/2)[phone]/1280000000*10^(1/2)*arcsin(20/11* 
x+1/11)*((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 = 82, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=\frac {1}{192000000} \, {\left (691200000 \, x^{5} + 338688000 \, x^{4} - 729302400 \, x^{3} - 233839520 \, x^{2} + 334643860 \, x + 39899709\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {368012183}{1280000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

1/192000000*(691200000*x^5 + 338688000*x^4 - 729302400*x^3 - 233839520*x^2 
 + 334643860*x + 39899709)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 368012183/128000 
0000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) 
/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=\frac {18}{5} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + \frac {441}{250} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {75969}{20000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {1461497}{1200000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {16732193}{9600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {368012183}{1280000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13299903}{64000000} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

18/5*sqrt(-10*x^2 - x + 3)*x^5 + 441/250*sqrt(-10*x^2 - x + 3)*x^4 - 75969 
/20000*sqrt(-10*x^2 - x + 3)*x^3 - 1461497/1200000*sqrt(-10*x^2 - x + 3)*x 
^2 + 16732193/9600000*sqrt(-10*x^2 - x + 3)*x - 368012183/1280000000*sqrt( 
10)*arcsin(-20/11*x - 1/11) + 13299903/64000000*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (115) = 230\).

Time = 0.17 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=\frac {9}{3200000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{80000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{640000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {29}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {4}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \] Input:

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

Output:

9/3200000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x 
 + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)* 
sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 
 9/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) 
 - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*s 
qrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 3/640000*sqrt(5)*(2*(4*(8*(6 
0*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 
 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 29/60000*sqrt( 
5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 478 
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/500*sqrt(5)*(2*(20*x - 
23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt( 
5*x + 3))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) 
+ 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx=-\frac {368012183 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{640000000}+\frac {18 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{5}+\frac {441 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{250}-\frac {75969 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{20000}-\frac {1461497 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1200000}+\frac {16732193 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{9600000}+\frac {13299903 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{64000000} \] Input:

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

Output:

( - 1104036549*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 691200 
0000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 + 3386880000*sqrt(5*x + 3)*sqrt( 
- 2*x + 1)*x**4 - 7293024000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 2338395 
200*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 3346438600*sqrt(5*x + 3)*sqrt( - 
 2*x + 1)*x + 398997090*sqrt(5*x + 3)*sqrt( - 2*x + 1))/1920000000