Integrand size = 26, antiderivative size = 216 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}} \]
-5787133/3072000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-526103/768000*(1-2*x)^(7/2)*( 3+5*x)^(5/2)-1/30*(1-2*x)^(7/2)*(2+3*x)^2*(3+5*x)^(7/2)-1/672000*(1-2*x)^( 7/2)*(3+5*x)^(7/2)*(245011+170940*x)+932023556783/65536000000*arcsin(1/11* 22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7702674023/1966080000*(1-2*x)^(3/2)*(3+5* x)^(1/2)+700243093/491520000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-63658463/16384000 *(1-2*x)^(7/2)*(3+5*x)^(1/2)+84729414253/6553600000*(1-2*x)^(1/2)*(3+5*x)^ (1/2)
Time = 0.33 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.48 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-4490138164311+11708962285875 x+81324587821180 x^2+65091129546400 x^3-209312726736000 x^4-440233726720000 x^5-52760857600000 x^6+605463552000000 x^7+636088320000000 x^8+206438400000000 x^9\right )-19572494692443 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1376256000000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-4490138164311 + 11708962285875*x + 81324587821180*x^2 + 65091129546400*x^3 - 209312726736000*x^4 - 440233726720000*x^5 - 5276085 7600000*x^6 + 605463552000000*x^7 + 636088320000000*x^8 + 206438400000000* x^9) - 19572494692443*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x] ])/(1376256000000*Sqrt[3 + 5*x])
Time = 0.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {111, 27, 164, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^3 (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{90} \int -\frac {3}{2} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{5/2} (407 x+262)dx-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \int (1-2 x)^{5/2} (3 x+2) (5 x+3)^{5/2} (407 x+262)dx-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \int (1-2 x)^{5/2} (5 x+3)^{5/2}dx}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{60} \left (\frac {1578309 \left (\frac {55}{24} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{12} (1-2 x)^{7/2} (5 x+3)^{5/2}\right )}{3200}-\frac {(1-2 x)^{7/2} (5 x+3)^{7/2} (170940 x+245011)}{11200}\right )-\frac {1}{30} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{7/2}\) |
-1/30*((1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2)) + (-1/11200*((1 - 2*x) ^(7/2)*(3 + 5*x)^(7/2)*(245011 + 170940*x)) + (1578309*(-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)*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]*Sqr t[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/ 6))/16))/20))/24))/3200)/60
3.25.11.3.1 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_))^(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]
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]
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 = 1.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(-\frac {\left (41287680000000 x^{8}+102445056000000 x^{7}+59625676800000 x^{6}-46327577600000 x^{5}-60250198784000 x^{4}-5712426076800 x^{3}+16445681555360 x^{2}+6397508631020 x -1496712721437\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{137625600000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {932023556783 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{131072000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(128\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (825753600000000 \sqrt {-10 x^{2}-x +3}\, x^{8}+2048901120000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+1192513536000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-926551552000000 x^{5} \sqrt {-10 x^{2}-x +3}-1205003975680000 x^{4} \sqrt {-10 x^{2}-x +3}-114248521536000 x^{3} \sqrt {-10 x^{2}-x +3}+328913631107200 x^{2} \sqrt {-10 x^{2}-x +3}+19572494692443 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+127950172620400 x \sqrt {-10 x^{2}-x +3}-29934254428740 \sqrt {-10 x^{2}-x +3}\right )}{2752512000000 \sqrt {-10 x^{2}-x +3}}\) | \(189\) |
-1/137625600000*(41287680000000*x^8+102445056000000*x^7+59625676800000*x^6 -46327577600000*x^5-60250198784000*x^4-5712426076800*x^3+16445681555360*x^ 2+6397508631020*x-1496712721437)*(-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)+932023556783/131072000000*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 = 97, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\frac {1}{137625600000} \, {\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {932023556783}{131072000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/137625600000*(41287680000000*x^8 + 102445056000000*x^7 + 59625676800000* x^6 - 46327577600000*x^5 - 60250198784000*x^4 - 5712426076800*x^3 + 164456 81555360*x^2 + 6397508631020*x - 1496712721437)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 932023556783/131072000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sq rt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.67 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=-\frac {3}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} - \frac {1047}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {111537}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {526103}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {526103}{7680000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {63658463}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {63658463}{245760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7702674023}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {932023556783}{131072000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7702674023}{6553600000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-3/10*(-10*x^2 - x + 3)^(7/2)*x^2 - 1047/1600*(-10*x^2 - x + 3)^(7/2)*x - 111537/224000*(-10*x^2 - x + 3)^(7/2) + 526103/384000*(-10*x^2 - x + 3)^(5 /2)*x + 526103/7680000*(-10*x^2 - x + 3)^(5/2) + 63658463/12288000*(-10*x^ 2 - x + 3)^(3/2)*x + 63658463/245760000*(-10*x^2 - x + 3)^(3/2) + 77026740 23/327680000*sqrt(-10*x^2 - x + 3)*x - 932023556783/131072000000*sqrt(10)* arcsin(-20/11*x - 1/11) + 7702674023/6553600000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (159) = 318\).
Time = 0.43 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.02 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\text {Too large to display} \]
3/2293760000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(8*(28*(160*x - 779)*(5*x + 3) + 297993)*(5*x + 3) - 16954963)*(5*x + 3) + 311501761)*(5*x + 3) - 1539630 1917)*(5*x + 3) + 129214816941)*(5*x + 3) - 1465300159701)*(5*x + 3) + 596 7262275723)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10642307691015*sqrt(2)*arcsin( 1/11*sqrt(22)*sqrt(5*x + 3))) + 9/40960000000*sqrt(5)*(2*(4*(8*(4*(16*(4*( 24*(140*x - 599)*(5*x + 3) + 175163)*(5*x + 3) - 4295993)*(5*x + 3) + 2656 20213)*(5*x + 3) - 2676516549)*(5*x + 3) + 35390483373)*(5*x + 3) - 164483 997363)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 309625826895*sqrt(2)*arcsin(1/11*s qrt(22)*sqrt(5*x + 3))) + 2217/358400000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(1 20*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)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 121/38400 000000*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))) - 15709 /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*sq rt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1429/1920000*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))) + 129/2000...
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2} \,d x \]