Integrand size = 26, antiderivative size = 179 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\frac {1152712429 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000000}+\frac {104792039 (1-2 x)^{3/2} \sqrt {3+5 x}}{384000000}+\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {12679836719 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000000 \sqrt {10}} \]
12679836719/12800000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+10479 2039/384000000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+9526549/96000000*(1-2*x)^(5/2)* (3+5*x)^(1/2)-271/2800*(1-2*x)^(7/2)*(2+3*x)^2*(3+5*x)^(1/2)-3/70*(1-2*x)^ (7/2)*(2+3*x)^3*(3+5*x)^(1/2)-1/22400000*(1-2*x)^(7/2)*(12923401+11603280* x)*(3+5*x)^(1/2)+1152712429/1280000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\frac {10 \sqrt {1-2 x} \left (-2761931223+291878173875 x+563652783740 x^2-711260264800 x^3-1822308048000 x^4+195575040000 x^5+2305152000000 x^6+1244160000000 x^7\right )-266276571099 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{268800000000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-2761931223 + 291878173875*x + 563652783740*x^2 - 71126 0264800*x^3 - 1822308048000*x^4 + 195575040000*x^5 + 2305152000000*x^6 + 1 244160000000*x^7) - 266276571099*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sq rt[3 + 5*x]])/(268800000000*Sqrt[3 + 5*x])
Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {111, 27, 170, 27, 164, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^4}{\sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{70} \int -\frac {(1-2 x)^{5/2} (3 x+2)^2 (813 x+500)}{2 \sqrt {5 x+3}}dx-\frac {3}{70} (3 x+2)^3 \sqrt {5 x+3} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{140} \int \frac {(1-2 x)^{5/2} (3 x+2)^2 (813 x+500)}{\sqrt {5 x+3}}dx-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{140} \left (-\frac {1}{60} \int -\frac {3 (1-2 x)^{5/2} (3 x+2) (48347 x+29702)}{2 \sqrt {5 x+3}}dx-\frac {271}{20} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \int \frac {(1-2 x)^{5/2} (3 x+2) (48347 x+29702)}{\sqrt {5 x+3}}dx-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{140} \left (\frac {1}{40} \left (\frac {66685843 \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )}{8000}-\frac {(1-2 x)^{7/2} \sqrt {5 x+3} (11603280 x+12923401)}{4000}\right )-\frac {271}{20} (1-2 x)^{7/2} (3 x+2)^2 \sqrt {5 x+3}\right )-\frac {3}{70} (1-2 x)^{7/2} (3 x+2)^3 \sqrt {5 x+3}\) |
(-3*(1 - 2*x)^(7/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/70 + ((-271*(1 - 2*x)^(7/2) *(2 + 3*x)^2*Sqrt[3 + 5*x])/20 + (-1/4000*((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]*( 12923401 + 11603280*x)) + (66685843*(((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))/8000) /40)/140
3.25.24.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {\left (248832000000 x^{6}+311731200000 x^{5}-147923712000 x^{4}-275707382400 x^{3}+23172376480 x^{2}+98827130860 x -920643741\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{26880000000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {12679836719 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{25600000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+6234624000000 x^{5} \sqrt {-10 x^{2}-x +3}-2958474240000 x^{4} \sqrt {-10 x^{2}-x +3}-5514147648000 x^{3} \sqrt {-10 x^{2}-x +3}+463447529600 x^{2} \sqrt {-10 x^{2}-x +3}+266276571099 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1976542617200 x \sqrt {-10 x^{2}-x +3}-18412874820 \sqrt {-10 x^{2}-x +3}\right )}{537600000000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
-1/26880000000*(248832000000*x^6+311731200000*x^5-147923712000*x^4-2757073 82400*x^3+23172376480*x^2+98827130860*x-920643741)*(-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)+1267983671 9/25600000000*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 = 0.49 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\frac {1}{26880000000} \, {\left (248832000000 \, x^{6} + 311731200000 \, x^{5} - 147923712000 \, x^{4} - 275707382400 \, x^{3} + 23172376480 \, x^{2} + 98827130860 \, x - 920643741\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {12679836719}{25600000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/26880000000*(248832000000*x^6 + 311731200000*x^5 - 147923712000*x^4 - 27 5707382400*x^3 + 23172376480*x^2 + 98827130860*x - 920643741)*sqrt(5*x + 3 )*sqrt(-2*x + 1) - 12679836719/25600000000*sqrt(10)*arctan(1/20*sqrt(10)*( 20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}{\sqrt {5 x + 3}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\frac {324}{35} \, \sqrt {-10 \, x^{2} - x + 3} x^{6} + \frac {4059}{350} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} - \frac {192609}{35000} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {28719519}{2800000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + \frac {144827353}{168000000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {4941356543}{1344000000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {12679836719}{25600000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {306881247}{8960000000} \, \sqrt {-10 \, x^{2} - x + 3} \]
324/35*sqrt(-10*x^2 - x + 3)*x^6 + 4059/350*sqrt(-10*x^2 - x + 3)*x^5 - 19 2609/35000*sqrt(-10*x^2 - x + 3)*x^4 - 28719519/2800000*sqrt(-10*x^2 - x + 3)*x^3 + 144827353/168000000*sqrt(-10*x^2 - x + 3)*x^2 + 4941356543/13440 00000*sqrt(-10*x^2 - x + 3)*x - 12679836719/25600000000*sqrt(10)*arcsin(-2 0/11*x - 1/11) - 306881247/8960000000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (134) = 268\).
Time = 0.36 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.49 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\frac {27}{448000000000} \, \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 {9}{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 {27}{320000000} \, \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 {11}{400000} \, \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 {13}{15000} \, \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 {2}{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 {8}{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 )} \]
27/448000000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 9493 3)*(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)*ar csin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/640000000*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)*ar csin(1/11*sqrt(22)*sqrt(5*x + 3))) + 27/320000000*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))) - 11/400000*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*sq rt(22)*sqrt(5*x + 3))) - 13/15000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 12 93)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt (5*x + 3))) + 2/125*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))) + 8/25*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. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{\sqrt {5\,x+3}} \,d x \]