Integrand size = 26, antiderivative size = 187 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {100451901 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}+\frac {3043997 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}+\frac {276727 (1-2 x)^{5/2} \sqrt {3+5 x}}{1280000}-\frac {75471 (1-2 x)^{7/2} \sqrt {3+5 x}}{128000}-\frac {2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac {263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac {1104970911 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200000 \sqrt {10}} \]
-2287/8000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-263/2800*(1-2*x)^(7/2)*(3+5*x)^(5/2 )-3/70*(1-2*x)^(7/2)*(2+3*x)*(3+5*x)^(5/2)+1104970911/512000000*arcsin(1/1 1*22^(1/2)*(3+5*x)^(1/2))*10^(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)+100451901/51200000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-313262829+8460686625 x+18845312020 x^2-19694810400 x^3-61957104000 x^4-282880000 x^5+78208000000 x^6+46080000000 x^7\right )-7734796377 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3584000000 \sqrt {3+5 x}} \]
(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])
Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16, 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 |
\(\displaystyle \frac {1}{140} \left (\frac {16009}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {263}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{5/2}\) |
(-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
3.24.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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]
Time = 1.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.63
method | result | size |
risch | \(-\frac {\left (9216000000 x^{6}+10112000000 x^{5}-6123776000 x^{4}-8717155200 x^{3}+1291331040 x^{2}+2994263780 x -104420943\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{358400000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1104970911 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (184320000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+202240000000 x^{5} \sqrt {-10 x^{2}-x +3}-122475520000 x^{4} \sqrt {-10 x^{2}-x +3}-174343104000 x^{3} \sqrt {-10 x^{2}-x +3}+25826620800 x^{2} \sqrt {-10 x^{2}-x +3}+7734796377 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+59885275600 x \sqrt {-10 x^{2}-x +3}-2088418860 \sqrt {-10 x^{2}-x +3}\right )}{7168000000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
-1/358400000*(9216000000*x^6+10112000000*x^5-6123776000*x^4-8717155200*x^3 +1291331040*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)+1104970911/1024000000 *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)
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.47 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {1}{358400000} \, {\left (9216000000 \, x^{6} + 10112000000 \, x^{5} - 6123776000 \, x^{4} - 8717155200 \, x^{3} + 1291331040 \, x^{2} + 2994263780 \, x - 104420943\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1104970911}{1024000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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))
\[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.62 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {9}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {323}{1400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {9141}{140000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {25157}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {25157}{640000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {9131991}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1104970911}{1024000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {9131991}{51200000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (136) = 272\).
Time = 0.42 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.39 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {3}{17920000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {23}{640000000} \, \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 {109}{960000000} \, \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 {341}{4800000} \, \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 {227}{120000} \, \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 {21}{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 {18}{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 )} \]
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))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2} \,d x \]