Integrand size = 26, antiderivative size = 149 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=-\frac {661 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac {20}{81} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {19573 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4536 \sqrt {7}} \]
-1/9*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-19573/31752*arctan(1/7*(1-2*x)^ (1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+20/81*arcsin(1/11*22^(1/2)*(3+5*x)^(1 /2))*10^(1/2)+37/36*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-661/1512*(1-2*x) ^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (6176+21762 x+19041 x^2\right )}{(2+3 x)^3}-7840 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-19573 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{31752} \]
((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(6176 + 21762*x + 19041*x^2))/(2 + 3*x)^3 - 7840*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 19573*Sqrt[7]*Arc Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/31752
Time = 0.24 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{9} \int -\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{2 (3 x+2)^3}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{(3 x+2)^3}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{6} \int \frac {\sqrt {5 x+3} (160 x+327)}{2 \sqrt {1-2 x} (3 x+2)^2}dx+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \int \frac {\sqrt {5 x+3} (160 x+327)}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{21} \int \frac {11200 x+13991}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \int \frac {11200 x+13991}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {11200}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {19573}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {19573}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {4480}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {39146}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {4480}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {4480}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {39146 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {2240}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {39146 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
-1/9*((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^3 + ((37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(6*(2 + 3*x)^2) + ((-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*( 2 + 3*x)) + ((2240*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (39146*A rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/42)/12)/6
3.24.44.3.1 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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 = 137, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (19041 x^{2}+21762 x +6176\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1512 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {10 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{81}+\frac {19573 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{63504}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(137\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (528471 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+211680 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+1056942 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+423360 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+704628 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +282240 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +799722 x^{2} \sqrt {-10 x^{2}-x +3}+156584 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+62720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+914004 x \sqrt {-10 x^{2}-x +3}+259392 \sqrt {-10 x^{2}-x +3}\right )}{63504 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(253\) |
-1/1512*(-1+2*x)*(3+5*x)^(1/2)*(19041*x^2+21762*x+6176)/(2+3*x)^3/(-(-1+2* x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+(10/81*10^(1/2)*ar csin(20/11*x+1/11)+19573/63504*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 = 0.23 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=-\frac {19573 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 7840 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (19041 \, x^{2} + 21762 \, x + 6176\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{63504 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
-1/63504*(19573*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*( 37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 7840*sqrt(10)* (27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3) *sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(19041*x^2 + 21762*x + 6176)*sqrt(5 *x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {4045}{1764} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {10}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {19573}{63504} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {8573}{10584} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {83 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1176 \, {\left (3 \, x + 2\right )}} \]
185/882*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54 *x^2 + 36*x + 8) + 37/196*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 404 5/1764*sqrt(-10*x^2 - x + 3)*x + 10/81*sqrt(10)*arcsin(20/11*x + 1/11) + 1 9573/63504*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 857 3/10584*sqrt(-10*x^2 - x + 3) + 83/1176*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (113) = 226\).
Time = 0.50 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.53 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {19573}{635040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {10}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (661 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 499520 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {139630400 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {558521600 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{756 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
19573/635040*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)))) + 10/81*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)))) - 11/756*sqrt(10)*(661*((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)) )^5 + 499520*((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)))^3 - 139630400*(sqrt(2)*sqrt (-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 558521600*sqrt(5*x + 3)/(sqrt(2)*s qrt(-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)^3
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^4} \,d x \]