Integrand size = 26, antiderivative size = 180 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {16985 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}-\frac {279015 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}} \]
-279015/1075648*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11 /7*(3+5*x)^(1/2)/(2+3*x)^4/(1-2*x)^(1/2)-131/196*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)/(2+3*x)^4-89/392*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-745/10976*(1-2*x )^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+16985/153664*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)
Time = 2.99 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {5 \left (-\frac {7 \sqrt {3+5 x} \left (-163152-538276 x-60048 x^2+1188045 x^3+917190 x^4\right )}{5 \sqrt {1-2 x} (2+3 x)^4}+55803 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+55803 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{1075648} \]
(5*((-7*Sqrt[3 + 5*x]*(-163152 - 538276*x - 60048*x^2 + 1188045*x^3 + 9171 90*x^4))/(5*Sqrt[1 - 2*x]*(2 + 3*x)^4) + 55803*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 55803*Sqrt[7 ]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x] ))]))/1075648
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, 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, 168, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{3/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {1}{7} \int -\frac {1145 x+676}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {1145 x+676}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{28} \int \frac {3 (2620 x+1539)}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \int \frac {2620 x+1539}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {1}{21} \int \frac {35 (712 x+425)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \int \frac {712 x+425}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {2980 x+3119}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {2980 x+3119}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {55803}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3397 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {55803}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {3397 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {55803}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {3397 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {3}{28} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {3397 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {55803 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {149 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^4}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + ((-131*Sqrt[1 - 2*x]*Sq rt[3 + 5*x])/(14*(2 + 3*x)^4) + (3*((-89*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*( 2 + 3*x)^3) + (5*((-149*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (( 3397*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (55803*ArcTan[Sqrt[1 - 2 *x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/28)/14
3.26.38.3.1 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_)^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. \(2(141)=282\).
Time = 3.97 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {\left (45200430 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+97934265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+60267240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+12840660 x^{4} \sqrt {-10 x^{2}-x +3}-6696360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+16632630 x^{3} \sqrt {-10 x^{2}-x +3}-17856960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -840672 x^{2} \sqrt {-10 x^{2}-x +3}-4464240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7535864 x \sqrt {-10 x^{2}-x +3}-2284128 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{2151296 \left (2+3 x \right )^{4} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
1/2151296*(45200430*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))*x^5+97934265*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^4+60267240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^3+12840660*x^4*(-10*x^2-x+3)^(1/2)-6696360*7^(1/2)*arctan(1/14*(37*x+20 )*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+16632630*x^3*(-10*x^2-x+3)^(1/2)-178569 60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-840672*x^2 *(-10*x^2-x+3)^(1/2)-4464240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^ 2-x+3)^(1/2))-7535864*x*(-10*x^2-x+3)^(1/2)-2284128*(-10*x^2-x+3)^(1/2))*( 1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=-\frac {279015 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (917190 \, x^{4} + 1188045 \, x^{3} - 60048 \, x^{2} - 538276 \, x - 163152\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
-1/2151296*(279015*sqrt(7)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(917190*x^4 + 1188045*x^3 - 60048*x^2 - 538276*x - 163152)*s qrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (141) = 282\).
Time = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.64 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {279015}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {84925 \, x}{230496 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {131015}{460992 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{252 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {169}{3528 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {649}{4704 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {2475}{21952 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
279015/2151296*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 84925/230496*x/sqrt(-10*x^2 - x + 3) + 131015/460992/sqrt(-10*x^2 - x + 3 ) - 1/252/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x ^2 - x + 3)) + 169/3528/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 649/4 704/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10 *x^2 - x + 3)) - 2475/21952/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (141) = 282\).
Time = 0.69 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.19 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {55803}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {176 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (178579 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 183436680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 17824632000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2829942080000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {11319768320000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
55803/4302592*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(-1 0*x + 5) - sqrt(22)))) - 176/84035*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/( 2*x - 1) - 11/537824*sqrt(10)*(178579*((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)))^7 + 183436680*((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 - 17824632000*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))^3 - 2829942080000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) + 11319768320000*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)^4
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5} \,d x \]