\(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx\) [1004]

Optimal result

Integrand size = 26, antiderivative size = 182 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {100451901 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}+\frac {3043997 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}+\frac {276727 (1-2 x)^{5/2} \sqrt {3+5 x}}{1280000}-\frac {75471 (1-2 x)^{7/2} \sqrt {3+5 x}}{128000}-\frac {2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac {683 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}+\frac {9}{140} (1-2 x)^{9/2} (3+5 x)^{5/2}+\frac {1104970911 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200000 \sqrt {10}} \] Output:

100451901/51200000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+3043997/5120000*(1-2*x)^(3/ 
2)*(3+5*x)^(1/2)+276727/1280000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-75471/128000*( 
1-2*x)^(7/2)*(3+5*x)^(1/2)-2287/8000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-683/2800* 
(1-2*x)^(7/2)*(3+5*x)^(5/2)+9/140*(1-2*x)^(9/2)*(3+5*x)^(5/2)[phone]/5 
12000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-313262829+8460686625 x+18845312020 x^2-19694810400 x^3-61957104000 x^4-282880000 x^5+78208000000 x^6+46080000000 x^7\right )-7734796377 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3584000000 \sqrt {3+5 x}} \] Input:

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

Output:

(10*Sqrt[1 - 2*x]*(-313262829 + 8460686625*x + 18845312020*x^2 - 196948104 
00*x^3 - 61957104000*x^4 - 282880000*x^5 + 78208000000*x^6 + 46080000000*x 
^7) - 7734796377*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(3 
584000000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {101, 27, 90, 60, 60, 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.

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{70} \int -\frac {1}{2} (1-2 x)^{5/2} (5 x+3)^{3/2} (789 x+512)dx-\frac {3}{70} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{140} \int (1-2 x)^{5/2} (5 x+3)^{3/2} (789 x+512)dx-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\)

Input:

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

Output:

(-3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/70 + ((-263*(1 - 2*x)^(7/2) 
*(3 + 5*x)^(5/2))/20 + (16009*(-1/10*((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)) + ( 
33*(-1/8*((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]) + (11*(((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))/16))/20))/40)/140
 

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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 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 = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.65

method	result	size
risch	\(-\frac {\left (9216000000 x^{6}[phone] x^{5}-6123776000 x^{4}-8717155200 x^{3}[phone] x^{2}[phone] x -104420943\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{358400000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1104970911 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (184320000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-122475520000 x^{4} \sqrt {-10 x^{2}-x +3}-174343104000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-2088418860 \sqrt {-10 x^{2}-x +3}\right )}{7168000000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

Input:

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

Output:

-1/358400000*(9216000000*x^6+10112000000*x^5-6123776000*x^4-8717155200*x^3 
[phone]*x^2+2994263780*x-104420943)*(-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]/1024000000 
*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 = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {1}{358400000} \, {\left (9216000000 \, x^{6} + 10112000000 \, x^{5} - 6123776000 \, x^{4} - 8717155200 \, x^{3} + 1291331040 \, x^{2} + 2994263780 \, x - 104420943\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1104970911}{1024000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

1/358400000*(9216000000*x^6 + 10112000000*x^5 - 6123776000*x^4 - 871715520 
0*x^3 + 1291331040*x^2 + 2994263780*x - 104420943)*sqrt(5*x + 3)*sqrt(-2*x 
 + 1) - 1104970911/1024000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqr 
t(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {9}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {323}{1400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {9141}{140000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {25157}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {25157}{640000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {9131991}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1104970911}{1024000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {9131991}{51200000} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

9/35*(-10*x^2 - x + 3)^(5/2)*x^2 + 323/1400*(-10*x^2 - x + 3)^(5/2)*x - 91 
41/140000*(-10*x^2 - x + 3)^(5/2) + 25157/32000*(-10*x^2 - x + 3)^(3/2)*x 
+ 25157/640000*(-10*x^2 - x + 3)^(3/2) + 9131991/2560000*sqrt(-10*x^2 - x 
+ 3)*x - 1104970911/1024000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 9131991/ 
51200000*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (131) = 262\).

Time = 0.19 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

3/17920000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933) 
*(5*x + 3) - 7838433)*(5*x + 3) + 98794353)*(5*x + 3) - 1568443065)*(5*x + 
 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arcs 
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 23/640000000*sqrt(5)*(2*(4*(8*(4*(16*(1 
00*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)*arc 
sin(1/11*sqrt(22)*sqrt(5*x + 3))) + 109/960000000*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*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x 
 + 3))) - 341/4800000*sqrt(5)*(2*(4*(8*(60*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))) - 227/120000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) 
+ 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)* 
sqrt(5*x + 3))) + 21/500*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))) + 18/25*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. \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=-\frac {1104970911 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{512000000}+\frac {180 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{6}}{7}+\frac {395 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{14}-\frac {23921 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{1400}-\frac {2724111 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{112000}+\frac {8070819 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{2240000}+\frac {149713189 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{17920000}-\frac {104420943 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{358400000} \] Input:

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

Output:

( - 7734796377*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 921600 
00000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**6 + 101120000000*sqrt(5*x + 3)*sqr 
t( - 2*x + 1)*x**5 - 61237760000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 871 
71552000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 12913310400*sqrt(5*x + 3)*s 
qrt( - 2*x + 1)*x**2 + 29942637800*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 1044 
209430*sqrt(5*x + 3)*sqrt( - 2*x + 1))/3584000000