Integrand size = 26, antiderivative size = 209 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{327680000 \sqrt {10}} \]
-132451/153600*(1-2*x)^(7/2)*(3+5*x)^(3/2)-12041/38400*(1-2*x)^(7/2)*(3+5* x)^(5/2)-999/11200*(1-2*x)^(7/2)*(3+5*x)^(7/2)-3/80*(1-2*x)^(7/2)*(2+3*x)* (3+5*x)^(7/2)+21331366001/3276800000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*1 0^(1/2)+176292281/98304000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+16026571/24576000*( 1-2*x)^(5/2)*(3+5*x)^(1/2)-1456961/819200*(1-2*x)^(7/2)*(3+5*x)^(1/2)+1939 215091/327680000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.47 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-67244995017+395916406125 x+1604753427460 x^2+22475240800 x^3-5109421872000 x^4-4777381120000 x^5+4571417600000 x^6+9133056000000 x^7+3870720000000 x^8\right )-447958686021 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{68812800000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-67244995017 + 395916406125*x + 1604753427460*x^2 + 224 75240800*x^3 - 5109421872000*x^4 - 4777381120000*x^5 + 4571417600000*x^6 + 9133056000000*x^7 + 3870720000000*x^8) - 447958686021*Sqrt[30 + 50*x]*Arc Tan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(68812800000*Sqrt[3 + 5*x])
Time = 0.26 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {101, 27, 90, 60, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {1}{80} \int -\frac {1}{2} (1-2 x)^{5/2} (5 x+3)^{5/2} (999 x+652)dx-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{160} \int (1-2 x)^{5/2} (5 x+3)^{5/2} (999 x+652)dx-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \int (1-2 x)^{5/2} (5 x+3)^{5/2}dx-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{160} \left (\frac {12041}{20} \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )-\frac {999}{70} (1-2 x)^{7/2} (5 x+3)^{7/2}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{7/2}\) |
(-3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80 + ((-999*(1 - 2*x)^(7/2) *(3 + 5*x)^(7/2))/70 + (12041*(-1/12*((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2)) + ( 55*(-1/10*((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)) + (33*(-1/8*((1 - 2*x)^(7/2)*S qrt[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]*Sqrt[3 + 5*x])/5 + (11*ArcSi n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/20))/24))/20)/16 0
3.25.12.3.1 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 = 3.84 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(-\frac {\left (774144000000 x^{7}+1362124800000 x^{6}+97008640000 x^{5}-1013681408000 x^{4}-413675529600 x^{3}+252700365920 x^{2}+169330465940 x -22414998339\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6881280000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {21331366001 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6553600000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+27242496000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1940172800000 x^{5} \sqrt {-10 x^{2}-x +3}-20273628160000 x^{4} \sqrt {-10 x^{2}-x +3}-8273510592000 x^{3} \sqrt {-10 x^{2}-x +3}+5054007318400 x^{2} \sqrt {-10 x^{2}-x +3}+447958686021 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3386609318800 x \sqrt {-10 x^{2}-x +3}-448299966780 \sqrt {-10 x^{2}-x +3}\right )}{137625600000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
-1/6881280000*(774144000000*x^7+1362124800000*x^6+97008640000*x^5-10136814 08000*x^4-413675529600*x^3+252700365920*x^2+169330465940*x-22414998339)*(- 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)+21331366001/6553600000*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 = 92, normalized size of antiderivative = 0.44 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {1}{6881280000} \, {\left (774144000000 \, x^{7} + 1362124800000 \, x^{6} + 97008640000 \, x^{5} - 1013681408000 \, x^{4} - 413675529600 \, x^{3} + 252700365920 \, x^{2} + 169330465940 \, x - 22414998339\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {21331366001}{6553600000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/6881280000*(774144000000*x^7 + 1362124800000*x^6 + 97008640000*x^5 - 101 3681408000*x^4 - 413675529600*x^3 + 252700365920*x^2 + 169330465940*x - 22 414998339)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21331366001/6553600000*sqrt(10)* arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.61 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=-\frac {9}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {1839}{11200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {12041}{19200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {12041}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {1456961}{614400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1456961}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {176292281}{16384000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21331366001}{6553600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {176292281}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/80*(-10*x^2 - x + 3)^(7/2)*x - 1839/11200*(-10*x^2 - x + 3)^(7/2) + 120 41/19200*(-10*x^2 - x + 3)^(5/2)*x + 12041/384000*(-10*x^2 - x + 3)^(5/2) + 1456961/614400*(-10*x^2 - x + 3)^(3/2)*x + 1456961/12288000*(-10*x^2 - x + 3)^(3/2) + 176292281/16384000*sqrt(-10*x^2 - x + 3)*x - 21331366001/655 3600000*sqrt(10)*arcsin(-20/11*x - 1/11) + 176292281/327680000*sqrt(-10*x^ 2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (152) = 304\).
Time = 0.41 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.61 \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {3}{114688000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{560000000} \, \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 {937}{7680000000} \, \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 {3083}{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 {3181}{9600000} \, \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 {87}{40000} \, \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 {27}{125} \, \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 {54}{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 )} \]
3/114688000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(24*(140*x - 599)*(5*x + 3) + 17 5163)*(5*x + 3) - 4295993)*(5*x + 3) + 265620213)*(5*x + 3) - 2676516549)* (5*x + 3) + 35390483373)*(5*x + 3) - 164483997363)*sqrt(5*x + 3)*sqrt(-10* x + 5) - 309625826895*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/560 000000*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) + 8 438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arcsin(1/11 *sqrt(22)*sqrt(5*x + 3))) + 937/7680000000*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))) - 3083/960000000*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 ))) - 3181/9600000*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*sqr t(22)*sqrt(5*x + 3))) - 87/40000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 129 3)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt( 5*x + 3))) + 27/125*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))) + 54/25*sqrt(5)*(11*sqrt (2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + ...
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2} \,d x \]