Integrand size = 26, antiderivative size = 157 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=\frac {24251 \sqrt {1-2 x} \sqrt {3+5 x}}{3240}-\frac {247}{270} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}+\frac {326717 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9720 \sqrt {10}}+\frac {805}{243} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
-8/27*(1-2*x)^(3/2)*(3+5*x)^(3/2)-1/3*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)+ 805/243*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+326717/972 00*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-247/270*(3+5*x)^(3/2)*(1-2 *x)^(1/2)+24251/3240*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=\frac {\frac {30 \sqrt {1-2 x} \left (65154+160421 x+45165 x^2-47100 x^3+36000 x^4\right )}{(2+3 x) \sqrt {3+5 x}}-326717 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+322000 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{97200} \]
((30*Sqrt[1 - 2*x]*(65154 + 160421*x + 45165*x^2 - 47100*x^3 + 36000*x^4)) /((2 + 3*x)*Sqrt[3 + 5*x]) - 326717*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 322000*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9 7200
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{3} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{3 x+2}dx-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{45} \int \frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (494 x+61)}{3 x+2}dx+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (494 x+61)}{3 x+2}dx+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \int -\frac {(2616-24251 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {1}{30} \int \frac {(2616-24251 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{6} \int -\frac {326717 x+142678}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (-\frac {1}{12} \int \frac {326717 x+142678}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{12} \left (\frac {225400}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {326717}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{12} \left (\frac {225400}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {653434}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{12} \left (\frac {450800}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {653434}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{12} \left (-\frac {653434}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {64400}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{15} \left (\frac {1}{30} \left (\frac {1}{12} \left (-\frac {326717}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {64400}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {24251}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {247}{15} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {16}{45} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}\) |
-1/3*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x) - (5*((16*(1 - 2*x)^(3/2) *(3 + 5*x)^(3/2))/45 + ((247*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/15 + ((-24251* Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + ((-326717*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqr t[3 + 5*x]])/3 - (64400*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x ])])/3)/12)/30)/15))/6
3.25.4.3.1 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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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 = 1.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (7200 x^{3}-13740 x^{2}+17277 x +21718\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3240 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {326717 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{194400}+\frac {805 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{486}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(143\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (432000 x^{3} \sqrt {-10 x^{2}-x +3}+980151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -966000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -824400 x^{2} \sqrt {-10 x^{2}-x +3}+653434 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-644000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1036620 x \sqrt {-10 x^{2}-x +3}+1303080 \sqrt {-10 x^{2}-x +3}\right )}{194400 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(180\) |
-1/3240*(-1+2*x)*(3+5*x)^(1/2)*(7200*x^3-13740*x^2+17277*x+21718)/(2+3*x)/ (-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)-(-326717/1 94400*10^(1/2)*arcsin(20/11*x+1/11)+805/486*7^(1/2)*arctan(9/14*(20/3+37/3 *x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2)))*((1-2*x)*(3+5*x))^(1/2)/(1-2* x)^(1/2)/(3+5*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=\frac {322000 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 326717 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 60 \, {\left (7200 \, x^{3} - 13740 \, x^{2} + 17277 \, x + 21718\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{194400 \, {\left (3 \, x + 2\right )}} \]
1/194400*(322000*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5* x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 326717*sqrt(10)*(3*x + 2)*arctan (1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 60*(7200*x^3 - 13740*x^2 + 17277*x + 21718)*sqrt(5*x + 3)*sqrt(-2*x + 1)) /(3*x + 2)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=-\frac {2}{27} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {247}{54} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {326717}{194400} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {805}{486} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {15359}{3240} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{9 \, {\left (3 \, x + 2\right )}} \]
-2/27*(-10*x^2 - x + 3)^(3/2) - 247/54*sqrt(-10*x^2 - x + 3)*x + 326717/19 4400*sqrt(10)*arcsin(20/11*x + 1/11) - 805/486*sqrt(7)*arcsin(37/11*x/abs( 3*x + 2) + 20/11/abs(3*x + 2)) + 15359/3240*sqrt(-10*x^2 - x + 3) - 7/9*(- 10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (115) = 230\).
Time = 0.52 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.94 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=-\frac {161}{972} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{5400} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 151 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4817 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {326717}{194400} \, \sqrt {10} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{81 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-161/972*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*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)))) + 1/5400*(4*(8*sqrt(5)*(5*x + 3) - 151*sqrt(5))*(5*x + 3 ) + 4817*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 326717/194400*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)))) + 1078/81*sqrt(10)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2 )*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt (5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^2} \,d x \]