Integrand size = 26, antiderivative size = 173 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {465 \sqrt {3+5 x}}{9604 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {32 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^3}-\frac {23 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}-\frac {85 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {9395 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}} \]
-9395/134456*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11/21 *(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^3+465/9604*(3+5*x)^(1/2)/(1-2*x)^(1/2 )-32/147*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)-23/196*(3+5*x)^(1/2)/(2+3*x )^2/(1-2*x)^(1/2)-85/2744*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {-\frac {7 \sqrt {3+5 x} \left (-19296-80510 x-17127 x^2+193860 x^3+150660 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-28185 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{403368} \]
((-7*Sqrt[3 + 5*x]*(-19296 - 80510*x - 17127*x^2 + 193860*x^3 + 150660*x^4 ))/((1 - 2*x)^(3/2)*(2 + 3*x)^3) - 28185*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqr t[7]*Sqrt[3 + 5*x])])/403368
Time = 0.26 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 168, 27, 168, 27, 168, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{5/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {1}{21} \int -\frac {795 x+466}{2 (1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \int \frac {795 x+466}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{21} \int \frac {9 (640 x+373)}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \int \frac {640 x+373}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {1}{14} \int \frac {35 (184 x+117)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \int \frac {184 x+117}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {340 x+853}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {340 x+853}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {372 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {20669}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1879}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {372 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {3758}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {372 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{42} \left (\frac {3}{7} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {372 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}-\frac {3758 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {17 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {23 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {64 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + ((-64*Sqrt[3 + 5*x]) /(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (3*((-23*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]* (2 + 3*x)^2) + (5*((-17*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((372 *Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) - (3758*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq rt[3 + 5*x])])/(7*Sqrt[7]))/14))/4))/7)/42
3.26.97.3.1 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_))/((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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(134)=268\).
Time = 1.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {\left (3043980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+3043980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-1268325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-2109240 x^{4} \sqrt {-10 x^{2}-x +3}-1634730 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-2714040 x^{3} \sqrt {-10 x^{2}-x +3}+112740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +239778 x^{2} \sqrt {-10 x^{2}-x +3}+225480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1127140 x \sqrt {-10 x^{2}-x +3}+270144 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{806736 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
1/806736*(3043980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^5+3043980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^4-1268325*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3 -2109240*x^4*(-10*x^2-x+3)^(1/2)-1634730*7^(1/2)*arctan(1/14*(37*x+20)*7^( 1/2)/(-10*x^2-x+3)^(1/2))*x^2-2714040*x^3*(-10*x^2-x+3)^(1/2)+112740*7^(1/ 2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+239778*x^2*(-10*x^ 2-x+3)^(1/2)+225480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))+1127140*x*(-10*x^2-x+3)^(1/2)+270144*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {28185 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (150660 \, x^{4} + 193860 \, x^{3} - 17127 \, x^{2} - 80510 \, x - 19296\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{806736 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
-1/806736*(28185*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*a rctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(150660*x^4 + 193860*x^3 - 17127*x^2 - 80510*x - 19296)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.39 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {9395}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2325 \, x}{9604 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5395}{57624 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7625 \, x}{12348 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{567 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {169}{5292 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1987}{10584 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {2165}{222264 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
9395/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 23 25/9604*x/sqrt(-10*x^2 - x + 3) + 5395/57624/sqrt(-10*x^2 - x + 3) + 7625/ 12348*x/(-10*x^2 - x + 3)^(3/2) + 1/567/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) - 169/5292/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1987/10584/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 2165/222264/(-10*x^2 - x + 3)^ (3/2)
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (134) = 268\).
Time = 0.63 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.02 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {1879}{537824} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, {\left (512 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3201 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1260525 \, {\left (2 \, x - 1\right )}^{2}} - \frac {99 \, \sqrt {10} {\left (727 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 548800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {20776000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {83104000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
1879/537824*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)))) - 8/1260525*(512*sqrt(5)*(5*x + 3) - 3201*sqrt(5))*sq rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 99/67228*sqrt(10)*(727*((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 + 548800*((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 + 20776000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 83104000*sqr t(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) - s qrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \]