Integrand size = 26, antiderivative size = 150 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {15863 \sqrt {1-2 x} \sqrt {3+5 x}}{20736}-\frac {53}{192} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {23}{216} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {648919 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{62208 \sqrt {10}}+\frac {14}{243} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
1/12*(1-2*x)^(3/2)*(3+5*x)^(5/2)+14/243*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/( 3+5*x)^(1/2))*7^(1/2)+648919/622080*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10 ^(1/2)-53/192*(3+5*x)^(3/2)*(1-2*x)^(1/2)+23/216*(3+5*x)^(5/2)*(1-2*x)^(1/ 2)-15863/20736*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {30 \sqrt {1-2 x} \left (7167+187013 x+275940 x^2-285600 x^3-432000 x^4\right )-648919 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+35840 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{622080 \sqrt {3+5 x}} \]
(30*Sqrt[1 - 2*x]*(7167 + 187013*x + 275940*x^2 - 285600*x^3 - 432000*x^4) - 648919*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 35840*Sq rt[7]*Sqrt[3 + 5*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(622080 *Sqrt[3 + 5*x])
Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 175, 64, 104, 217, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 x+2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {1}{12} \int -\frac {\sqrt {1-2 x} (5 x+3)^{3/2} (115 x+58)}{2 (3 x+2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (115 x+58)}{3 x+2}dx+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{45} \int \frac {5 (5 x+3)^{3/2} (1431 x+170)}{2 \sqrt {1-2 x} (3 x+2)}dx+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \int \frac {(5 x+3)^{3/2} (1431 x+170)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (-\frac {1}{12} \int -\frac {3 \sqrt {5 x+3} (15863 x+12666)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \int \frac {\sqrt {5 x+3} (15863 x+12666)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (-\frac {1}{6} \int -\frac {648919 x+424250}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \int \frac {648919 x+424250}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {648919}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {25088}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1297838}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {25088}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1297838}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {50176}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1297838}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {7168}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{24} \left (\frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {648919}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {7168}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {15863}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {477}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {23}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}\) |
((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/12 + ((23*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)) /9 + ((-477*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 + ((-15863*Sqrt[1 - 2*x]*Sqrt [3 + 5*x])/6 + ((648919*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (7 168*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/12)/8)/18)/2 4
3.24.53.3.1 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[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 _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
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] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 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 = 132, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-5184000 x^{3} \sqrt {-10 x^{2}-x +3}-316800 x^{2} \sqrt {-10 x^{2}-x +3}+648919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-35840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3501360 x \sqrt {-10 x^{2}-x +3}+143340 \sqrt {-10 x^{2}-x +3}\right )}{1244160 \sqrt {-10 x^{2}-x +3}}\) | \(132\) |
| risch | \(\frac {\left (86400 x^{3}+5280 x^{2}-58356 x -2389\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{20736 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {648919 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1244160}-\frac {7 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{243}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(135\) |
1/1244160*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-5184000*x^3*(-10*x^2-x+3)^(1/2)-31 6800*x^2*(-10*x^2-x+3)^(1/2)+648919*10^(1/2)*arcsin(20/11*x+1/11)-35840*7^ (1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3501360*x*(-10*x^ 2-x+3)^(1/2)+143340*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {1}{20736} \, {\left (86400 \, x^{3} + 5280 \, x^{2} - 58356 \, x - 2389\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {7}{243} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {648919}{1244160} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/20736*(86400*x^3 + 5280*x^2 - 58356*x - 2389)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 7/243*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2* x + 1)/(10*x^2 + x - 3)) - 648919/1244160*sqrt(10)*arctan(1/20*sqrt(10)*(2 0*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{3 x + 2}\, dx \]
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {5}{12} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {7}{432} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2675}{1728} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {648919}{1244160} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{243} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3397}{20736} \, \sqrt {-10 \, x^{2} - x + 3} \]
5/12*(-10*x^2 - x + 3)^(3/2)*x - 7/432*(-10*x^2 - x + 3)^(3/2) + 2675/1728 *sqrt(-10*x^2 - x + 3)*x + 648919/1244160*sqrt(10)*arcsin(20/11*x + 1/11) - 7/243*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3397/2 0736*sqrt(-10*x^2 - x + 3)
Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.33 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {7}{2430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{518400} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 313 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2385 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 79315 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {648919}{1244160} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
-7/2430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1/518400*(12*(8*(36*sqrt(5)*(5*x + 3) - 313*sqrt(5))*(5 *x + 3) + 2385*sqrt(5))*(5*x + 3) + 79315*sqrt(5))*sqrt(5*x + 3)*sqrt(-10* x + 5) + 648919/1244160*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt( 2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{3\,x+2} \,d x \]