Integrand size = 26, antiderivative size = 194 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000000 \sqrt {10}} \]
-735439/1280000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-3/80*(1-2*x)^(7/2)*(2+3*x)^2*( 3+5*x)^(5/2)-9/448000*(1-2*x)^(7/2)*(3+5*x)^(5/2)*(18399+13480*x)+35532955 9167/81920000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+978869309/81 9200000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+88988119/204800000*(1-2*x)^(5/2)*(3+5* x)^(1/2)-24269487/20480000*(1-2*x)^(7/2)*(3+5*x)^(1/2)+32302687197/8192000 000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-346249385613+2250324002625 x+8626817121940 x^2-517857728800 x^3-27529736688000 x^4-23895957760000 x^5+25359744000000 x^6+47139840000000 x^7+19353600000000 x^8\right )-2487306914169 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{573440000000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-346249385613 + 2250324002625*x + 8626817121940*x^2 - 5 17857728800*x^3 - 27529736688000*x^4 - 23895957760000*x^5 + 25359744000000 *x^6 + 47139840000000*x^7 + 19353600000000*x^8) - 2487306914169*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(573440000000*Sqrt[3 + 5*x])
Time = 0.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {111, 27, 164, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^3 (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{80} \int -\frac {1}{2} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2} (1011 x+646)dx-\frac {3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{160} \int (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2} (1011 x+646)dx-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{160} \left (\frac {735439}{800} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {9 (1-2 x)^{7/2} (5 x+3)^{5/2} (13480 x+18399)}{2800}\right )-\frac {3}{80} (1-2 x)^{7/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
(-3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 + ((-9*(1 - 2*x)^(7/2) *(3 + 5*x)^(5/2)*(18399 + 13480*x))/2800 + (735439*(-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]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x] ])/(5*Sqrt[10])))/20))/6))/16))/20))/800)/160
3.24.99.3.1 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 = 1.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(-\frac {\left (3870720000000 x^{7}+7105536000000 x^{6}+808627200000 x^{5}-5264367872000 x^{4}-2347326614400 x^{3}+1304824422880 x^{2}+942468770660 x -115416461871\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{57344000000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {355329559167 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{163840000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (77414400000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+142110720000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+16172544000000 x^{5} \sqrt {-10 x^{2}-x +3}-105287357440000 x^{4} \sqrt {-10 x^{2}-x +3}-46946532288000 x^{3} \sqrt {-10 x^{2}-x +3}+26096488457600 x^{2} \sqrt {-10 x^{2}-x +3}+2487306914169 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+18849375413200 x \sqrt {-10 x^{2}-x +3}-2308329237420 \sqrt {-10 x^{2}-x +3}\right )}{1146880000000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
-1/57344000000*(3870720000000*x^7+7105536000000*x^6+808627200000*x^5-52643 67872000*x^4-2347326614400*x^3+1304824422880*x^2+942468770660*x-1154164618 71)*(-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)+355329559167/163840000000*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 = 0.47 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {1}{57344000000} \, {\left (3870720000000 \, x^{7} + 7105536000000 \, x^{6} + 808627200000 \, x^{5} - 5264367872000 \, x^{4} - 2347326614400 \, x^{3} + 1304824422880 \, x^{2} + 942468770660 \, x - 115416461871\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {355329559167}{163840000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/57344000000*(3870720000000*x^7 + 7105536000000*x^6 + 808627200000*x^5 - 5264367872000*x^4 - 2347326614400*x^3 + 1304824422880*x^2 + 942468770660*x - 115416461871)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 355329559167/163840000000* sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10* x^2 + x - 3))
\[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.69 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {27}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {6183}{5600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {71331}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {6491477}{22400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {8089829}{5120000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {8089829}{102400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2936607927}{409600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {355329559167}{163840000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {2936607927}{8192000000} \, \sqrt {-10 \, x^{2} - x + 3} \]
27/40*(-10*x^2 - x + 3)^(5/2)*x^3 + 6183/5600*(-10*x^2 - x + 3)^(5/2)*x^2 + 71331/224000*(-10*x^2 - x + 3)^(5/2)*x - 6491477/22400000*(-10*x^2 - x + 3)^(5/2) + 8089829/5120000*(-10*x^2 - x + 3)^(3/2)*x + 8089829/102400000* (-10*x^2 - x + 3)^(3/2) + 2936607927/409600000*sqrt(-10*x^2 - x + 3)*x - 3 55329559167/163840000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 2936607927/819 2000000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (143) = 286\).
Time = 0.41 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.81 \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {9}{573440000000} \, \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 {99}{89600000000} \, \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 {1029}{12800000000} \, \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 {457}{240000000} \, \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 {409}{1920000} \, \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 {101}{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 {69}{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 {36}{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 )} \]
9/573440000000*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))) + 99/89 600000000*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)*arcsin(1 /11*sqrt(22)*sqrt(5*x + 3))) + 1029/12800000000*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))) - 457/240000000*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))) - 409/1920000*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))) - 101/60000*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)*s qrt(5*x + 3))) + 69/500*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))) + 36/25*sqrt(5)*(11* sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*...
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2} \,d x \]