Integrand size = 26, antiderivative size = 180 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}-\frac {167155 \sqrt {1-2 x} \sqrt {3+5 x}}{1382976 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {168795 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}} \]
-168795/1075648*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11 /7*(3+5*x)^(3/2)/(2+3*x)^4/(1-2*x)^(1/2)+131/588*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)/(2+3*x)^4-3653/3528*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-38365/98784*( 1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2-167155/1382976*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {121 \left (\frac {7 \sqrt {3+5 x} \left (53136+687828 x+2184144 x^2+2578615 x^3+1002930 x^4\right )}{121 \sqrt {1-2 x} (2+3 x)^4}-1395 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{1075648} \]
(121*((7*Sqrt[3 + 5*x]*(53136 + 687828*x + 2184144*x^2 + 2578615*x^3 + 100 2930*x^4))/(121*Sqrt[1 - 2*x]*(2 + 3*x)^4) - 1395*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/1075648
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {1}{7} \int -\frac {\sqrt {5 x+3} (815 x+456)}{2 \sqrt {1-2 x} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {\sqrt {5 x+3} (815 x+456)}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \int \frac {106240 x+62303}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {1}{21} \int \frac {35 (29224 x+16925)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \int \frac {29224 x+16925}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {153460 x+91163}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {153460 x+91163}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {303831}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {33431 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {303831}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {33431 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {303831}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {33431 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{84} \left (\frac {5}{6} \left (\frac {1}{28} \left (-\frac {303831 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {33431 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {7673 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{42 (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + ((131*Sqrt[1 - 2*x]*S qrt[3 + 5*x])/(42*(2 + 3*x)^4) + ((-3653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*( 2 + 3*x)^3) + (5*((-7673*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ( (-33431*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (303831*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/84)/14
3.26.48.3.1 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*((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[((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 = 1.19 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {\left (27344790 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+59247045 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+36459720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-14041020 x^{4} \sqrt {-10 x^{2}-x +3}-4051080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-36100610 x^{3} \sqrt {-10 x^{2}-x +3}-10802880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -30578016 x^{2} \sqrt {-10 x^{2}-x +3}-2700720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-9629592 x \sqrt {-10 x^{2}-x +3}-743904 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{2151296 \left (2+3 x \right )^{4} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
1/2151296*(27344790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))*x^5+59247045*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^4+36459720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^3-14041020*x^4*(-10*x^2-x+3)^(1/2)-4051080*7^(1/2)*arctan(1/14*(37*x+20 )*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-36100610*x^3*(-10*x^2-x+3)^(1/2)-108028 80*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-30578016*x ^2*(-10*x^2-x+3)^(1/2)-2700720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10* x^2-x+3)^(1/2))-9629592*x*(-10*x^2-x+3)^(1/2)-743904*(-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 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=-\frac {168795 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (1002930 \, x^{4} + 2578615 \, x^{3} + 2184144 \, x^{2} + 687828 \, x + 53136\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
-1/2151296*(168795*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*(1002930*x^4 + 2578615*x^3 + 2184144*x^2 + 687828*x + 53136) *sqrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64* x - 16)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (141) = 282\).
Time = 0.31 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.64 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {168795}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {835775 \, x}{2074464 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {843155}{4148928 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{756 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {787}{31752 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {20681}{127008 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {69575}{197568 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
168795/2151296*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 835775/2074464*x/sqrt(-10*x^2 - x + 3) + 843155/4148928/sqrt(-10*x^2 - x + 3) + 1/756/(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(-1 0*x^2 - x + 3)) - 787/31752/(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)) + 2 0681/127008/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4* sqrt(-10*x^2 - x + 3)) - 69575/197568/(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 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {33759}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {968 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (10277 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 10598840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 3966648000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {122821440000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {491285760000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
33759/4302592*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)))) - 968/84035*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/( 2*x - 1) - 121/537824*sqrt(10)*(10277*((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 + 10598840*((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 + 3966648000*((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 + 122821440000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/ sqrt(5*x + 3) - 491285760000*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. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5} \,d x \]